Safety and Efficacy Study of Vintafolide and Vintafolide Plus Paclitaxel Compared to Paclitaxel Alone in Participants With Triple Negative Breast Cancer (TNBC) (MK-8109-004)

Inclusion Criteria: * Female with histologically or cytologically confirmed estrogen receptor negative, progesterone receptor negative, human epidermal growth factor receptor 2 (HER-2) negative advanced breast cancer that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective. * Has developed progressive disease following at least 1 (and not more than 4) prior chemotherapeutic regimens for breast cancer, which was administered for treatment of locally advanced, locally recurrent and/or metastatic disease. \[At least 1 regimen must have included a taxane (e.g., paclitaxel, docetaxel) in any combination or order.\] * Has at least a single measurable target lesion on a radiological evaluation that is conducted no more than 4 weeks prior to beginning of study drug. (Measurable lesions should not have received prior radiation therapy.) * Has not received chemotherapy for 4 weeks prior to the initiation of study drug and must have recovered to ≤Common Terminology Criteria for Adverse Events (CTCAE) grade 1 toxicities related to prior chemotherapies. * Female participants of childbearing potential should have a negative urine or serum pregnancy test within 72 hours prior to receiving the first dose of study drug. * Female participants of childbearing potential should be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 30 days after the last dose of study drug. * Has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. * Has no history of a prior malignancy with the exception of cervical intraepithelial neoplasia; basal cell carcinoma of the skin or has undergone potentially curative therapy with no evidence of that disease for 5 years. * Has adequate organ function. Exclusion Criteria: * Has had chemotherapy, radiotherapy, or biological therapy (including monoclonal antibodies) within 4 weeks prior to study drug administration or has not recovered (≤Grade 1 or baseline) from adverse experiences due to agents administered more than 4 weeks earlier. * Is currently participating or has participated in a study with an investigational compound or device within 28 days of initial dosing on this study. * Has received more than 4 prior cytotoxic regimens for metastatic disease. Adjuvant treatments would not count towards this criterion. * Has a primary central nervous system (CNS) tumor. * Has active CNS metastases and/or carcinomatous meningitis. * Has had prior therapy with vinorelbine (Navelbine®) or vinca-containing compounds. * Has known hypersensitivity to paclitaxel, docetaxel, other taxane therapies, drugs formulated with polyoxyethylated castor oil (Cremophor EL®), vinblastine, other vinca derived therapies, or their components or analogs. * Has pre-existing neuropathy \>Grade 2. * Has a recent (i.e., ≤6 weeks) history of abdominal surgery or peritonitis. * Has a bowel occlusion or sub-occlusion. * Has had prior abdominal or pelvic radiation therapy, or radiation therapy to \>10% of the bone marrow at any time in the past, or prior radiation therapy within the last 3 years to the breast/sternum, dermal lesions, head, or neck. * Requires anti-folate therapy. * Has known psychiatric or substance abuse disorders. * Is a known regular user (including "recreational use") of any illicit drugs or had a recent history (within the last year) of drug or alcohol abuse. * Is expecting to reproduce within the projected duration of the study, and women who are pregnant or breastfeeding. * Is known to be Human Immunodeficiency Virus (HIV)-positive. * Has known active Hepatitis B or C. * Has symptomatic ascites or pleural effusion. * Has had a prior stem cell or bone marrow transplant.