Patient description: The patient is a 37-year-old woman who came to the clinic for a routine Pap smear. The test revealed stage 1B of cervical cancer. The patient tested positive for HPV 16. She has three sexual partners and four children. She underwent tubal ligation. She smokes 30 packs/year and drinks alcohol frequently. She is otherwise healthy. She was offered a radical hysterectomy.

Clinical trial eligibility criteria: eligible ages (years): 18.0-999.0,  Advanced Breast Carcinoma Advanced Malignant Solid Neoplasm Advanced Prostate Carcinoma Anatomic Stage III Breast Cancer AJCC v8 Anatomic Stage IIIA Breast Cancer AJCC v8 Anatomic Stage IIIB Breast Cancer AJCC v8 Anatomic Stage IIIC Breast Cancer AJCC v8 Anatomic Stage IV Breast Cancer AJCC v8 Castration-Resistant Prostate Carcinoma Metastatic Breast Carcinoma Metastatic Malignant Solid Neoplasm Metastatic Prostate Carcinoma Prognostic Stage III Breast Cancer AJCC v8 Prognostic Stage IIIA Breast Cancer AJCC v8 Prognostic Stage IIIB Breast Cancer AJCC v8 Prognostic Stage IIIC Breast Cancer AJCC v8 Prognostic Stage IV Breast Cancer AJCC v8 Stage III Prostate Cancer AJCC v8 Stage IIIA Prostate Cancer AJCC v8 Stage IIIB Prostate Cancer AJCC v8 Stage IIIC Prostate Cancer AJCC v8 Stage IV Prostate Cancer AJCC v8 Stage IVA Prostate Cancer AJCC v8 Stage IVB Prostate Cancer AJCC v8 Triple-Negative Breast Carcinoma Unresectable Solid Neoplasm Patients must have histologically confirmed malignancy that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective Patients must be able to swallow and retain oral medications and be without gastrointestinal illnesses that would preclude absorption of AZD8186 Unlimited prior therapies allowed Docetaxel appropriate Patients who have not received prior docetaxel (or other taxane therapy) in the advanced setting are eligible for all cohorts Patients who have previously received docetaxel (or other taxane therapy) in the advanced setting are eligible for the dose escalation cohort only, if anticipated to have maintained taxane sensitivity and in the opinion of the investigator would still benefit from further docetaxel therapy Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%) Leukocytes >= 3,000/mcL Absolute neutrophil count >= 1,500/mcL Hemoglobin >= 8 g/L  HER2 positive breast cancer Prior treatment with PI3K/AKT inhibitors Any known concurrent RAF or PIK3CA mutation Patients who have had chemotherapy, radiotherapy, immunotherapy or anticancer agents within 4 weeks (6 weeks for nitrosoureas or mitomycin C) of the first dose of study treatment, except hormonal therapy with luteinizing hormone-releasing hormone (LHRH) analogues for medical castration in patients with prostate cancer and breast cancer, which are permitted Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities > grade 1) Patients who are receiving any other investigational agents Patients with known, untreated or unstable brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; patients with treated brain metastases are eligible if the metastases have been radiographically and clinically stable for at least one month; if on steroids for this indication, the patient must be on a stable dose for at least one month History of clinically significant allergic reactions attributed to compounds of similar chemical or biologic composition to AZD8186 or docetaxel or to docetaxel itself Patients receiving any medications or substances that are strong inhibitors and/or strong or moderate inducers of CYP3A4 are ineligible; because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product Existing bleeding or condition associated with increased risk of bleeding