Aurora Kinase Inhibitor LY3295668 in Combination With Osimertinib for the Treatment of Advanced or Metastatic EGFR-Mutant Non-squamous Non-small Cell Lung Cancer

Inclusion Criteria: 1. Written informed consent and HIPAA authorization for release of personal health information prior to registration. NOTE: HIPAA authorization may be included in the informed consent. 2. Age ≥ 18 years at the time of consent. 3. Histologically or cytologically confirmed non-squamous, non-small cell lung cancer 4. Locally advanced or metastatic disease. 5. Patients must have one of the following: * NSCLC which harbours EGFR Exon 19 deletion. * NSCLC which harbours EGFR L858R mutation. EGFR deletion/mutation must be documented by a Clinical Laboratory Improvement Amendments (CLIA) certified test (either from tissue or ctDNA from blood is allowed) 6. The patient must have received a third-generation EGFR TKI treatment, regardless of the T790M status. The patient must have progressed on osimertinib, or an equivalent third-generation EGFR TKI with regulatory approval. 7. Eastern Cooperative Oncology Group (ECOG) performance status 0-2 (Appendix A) 8. At least one lesion, not previously irradiated and not chosen for biopsy during the study screening period, that can be accurately measured at baseline at equal or greater than 10mm in the longest dimension by RECIST 1.1. 9. Ability to take pills or capsules by mouth 10. Prior treatment with cytotoxic chemotherapy or immunotherapy is allowed. Up to 3 lines of prior therapy is allowed. 11. Patients must have adequate hematologic, coagulation, hepatic, and renal function. All laboratory tests must be obtained less than 4 weeks from study entry. This includes: * ANC \>/= 1,500/mm3 * platelet count \>/=100,000/mm3 * HgB ≥ 9 g/dL * Creatinine ≤ 1.5x ULN or creatinine clearance (measured via 24-hour urine collection) ≥40 mL/minute (that is, if serum creatinine is \>1.5 times the ULN, a 24-hour urine collection to calculate creatinine clearance must be performed). * Total Serum Bilirubin ≤ 1.5 x ULN (Patients with known Gilbert Syndrome, a total bilirubin ≤ 3.0 x ULN, with direct bilirubin ≤ 1.5 x ULN) * SGOT, SGPT ≤ 3 X ULN if no liver metastasis present * SGOT, SGPT ≤ 5 X ULN if liver metastasis present 12. Females of childbearing potential must not be breast feeding and must have a negative serum or urine pregnancy test within 7 days of starting of treatment. The patient must agree to use adequate contraception for a minimum of two weeks prior to receiving study medication until 3 months after discontinuation of the study medication. Acceptable methods of contraception include total and true sexual abstinence, hormonal contraceptives that are not prone to drug-drug interactions (IUS Levonorgestrel Intra Uterine System (Mirena), Medroxyprogesterone injections (Depo-Provera)), copper-banded intra-uterine devices, and vasectomized partner. All hormonal methods of contraception should be used in combination with the use of a condom by their sexual male partner. Females of childbearing potential are defined as those who are not surgically sterile (i.e., bilateral tubal ligation, bilateral oophorectomy, or complete hysterectomy) or postmenopausal (defined as 12 months with no menses without an alternative medical cause). Women will be considered post-menopausal if they have been amenorrheic for the past 12 months without an alternative medical cause. The following age-specific requirements must also apply: Women \< 50 years old: they would be considered post-menopausal if they have been amenorrheic for the past 12 months or more following cessation of exogenous hormonal treatments. The levels of Luteinizing Hormone (LH) and Follicle-Stimulating Hormone (FSH) must also be in the post-menopausal range (as per the institution). Women ≥ 50 years old: they would be considered post-menopausal if they have been amenorrheic for the past 12 months or more following cessation of all exogenous hormonal treatments, or have had radiation-induced oophorectomy with the last menses \> 1 year ago, or have had chemotherapy-induced menopause with \>1 year interval since last menses, or have had surgical sterilization by either bilateral oophorectomy or hysterectomy. 13. Non-sterilized males who are sexually active with a female partner of childbearing potential must use adequate contraception for the duration of the study and 3 months after the last dose of study medication. Adequate contraception methods include: birth control pills (e.g. combined oral contraceptive pill), barrier protection (e.g. condom plus spermicide, cervical/vault cap or intrauterine device), and abstinence. Patients should not father a child for 6 months after completion of the study medication. Patients should refrain from donating sperm from the start of dosing until 6 months after discontinuing the study medication. If male patients wish to father children they should be advised to arrange for freezing of sperm samples prior to the start of the study medication. 14. Participants of childbearing potential who are sexually active and their partners must agree to abstain from heterosexual activity or to use 2 forms of effective methods of contraception. Non-sterilized males who are sexually active with a female partner of childbearing potential must use adequate contraception for the duration of the study and for 3 months after the last dose of study treatment. Patients should not father a child during the study or for 6 months after completion of study treatment. Patients should refrain from donating sperm from the start of dosing until 6 months after discontinuing the study treatment. Two contraception methods can be comprised of two barrier methods, or a barrier method plus a hormonal method. See below for options: Acceptable non-hormonal birth control methods: * Total sexual abstinence ie, refrain from any form of sexual intercourse in line with the patients' usual and/or preferred lifestyle. Abstinence must be for the total duration as described above. Periodic abstinence (eg, calendar ovulation, symptothermal, post-ovulation methods) and withdrawal are not acceptable methods of contraception. * Vasectomised sexual partner PLUS male condom. With participant assurance that partner received post-vasectomy confirmation of azoospermia. * Tubal occlusion PLUS male condom * Intrauterine Device PLUS male condom. Provided coils are copper-banded. Acceptable hormonal methods: * Etonogestrel implants (eg, Implanon®, Norplant®) PLUS male condom * Normal and low dose combined oral pills PLUS male condom * Hormonal shot or injection (eg, Depo-Provera) PLUS male condom * Intrauterine system device (eg, levonorgestrel-releasing intrauterine system - Mirena®) PLUS male condom Exclusion Criteria: 1. Previous treatment with other aurora kinase inhibitors. 2. Spinal cord compression or brain metastases unless asymptomatic or stable for at least 2 weeks prior to start of study treatment. 3. Any unresolved toxicities from prior therapy greater than CTCAE grade 1 (with the exception of alopecia grade 2) at the time of starting study treatment. 4. Any evidence of severe or uncontrolled systemic diseases. Screening for chronic conditions is not required. 5. Refractory nausea and vomiting, chronic gastrointestinal diseases, inability to swallow the formulated product or previous significant bowel resection that would preclude adequate absorption of osimertinib. 6. Males and females of reproductive potential who are not using and effective method of birth control and females who are pregnant or breastfeeding or have a positive (urine or serum) pregnancy test prior to study entry. 7. Judgment by the investigator that the patient should not participate in the study if the patient is unlikely to comply with study procedures, restrictions and requirement. 8. Any of the following cardiac criteria: * Mean resting corrected QT interval (QTc using Fridericia's formula) \> 470 msec, except in patients with existing bundle block, which artificially prolongs QTc. * Any clinically important abnormalities in rhythm, conduction or morphology of resting ECG e.g., complete left bundle branch block, third degree heart block, second degree heart block, PR interval \>250msec * Any factors that increase the risk of QTc prolongation or risk of arrhythmic events such as heart failure, hypokalemia, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years of age in first degree relatives or any concomitant medication known to prolong the QT interval. * The patient has experienced any arterial thromboembolic events, including but not limited to myocardial infarction, transient ischemic attack, cerebrovascular accident, or unstable angina, within 6 months prior to first dose of protocol therapy. * The patient has uncontrolled or poorly-controlled hypertension (\>160 mmHg systolic or \> 100 mmHg diastolic for \>4 weeks) despite standard medical management.