Nighttime Agitation and Restless Legs Syndrome in People With Alzheimer's Disease

Inclusion Criteria: * Aged \>=55 years * Clinical Dementia Rating (CDR) score of 0.5-3, indicating very mild to severe dementia * Physician diagnosis of dementia of the Alzheimer's type * Nighttime agitation, defined as Cohen Mansfield Agitation Inventory, Direct Observation total score \>=35 * Opinion of the participant's physician that medication for agitation is appropriate * RLS diagnosis by study advanced practice nurse (APN) or registered nurse (RN) (in consult with the participant's physician, and the investigators), using the Behavioral Indicators Test-Restless Legs * Medically stable, defined as unchanged medications within 14 days and the absence of fever or other signs and symptoms of acute illness or delirium (e.g. urinary tract infection, pneumonia) that may cause agitation or interfere with the study protocol * Able to swallow medication * Ambulatory, with and without assistance * If currently being treated for RLS, may be included if still having RLS symptoms/signs and confirmed as appropriate for inclusion by medical review Exclusion Criteria: * Received \>= 50 morphine milligram equivalents per day (MME/d) in the 14 days prior to the randomization decision, because morphine and GEn taken together have a higher incidence of sedation and dizziness than either drug alone * Currently being treated for RLS with gabapentin or GEn * Diagnosis of Parkinson's disease (PD) or any other disorder causing tremor because extrapyramidal symptoms may confound RLS diagnosis and actigraphy * Receiving gabapentin * Severe psychosis * Alcohol consumption because combining alcohol and GEn may increase sedation and other adverse events * Treatment with GEn is contraindicated, such as when a potential participant is receiving multiple antiepileptic drugs, in the opinion of the study APN or RN, participant's physician, or study medical team * Failure of past treatment with gabapentin or GEn * Compromised renal function as indicated by creatinine clearance \<15 or on hemodialysis * Current participation in a clinical trial or in any study that may affect study outcomes * Determined to be at risk for suicide by the study APN, RN, or participant's physician * Any condition, that in the opinion of the study APN or RN, participant's physician, or study medical team, makes it medically inappropriate for the patient to enroll in the trial * Persons living independently in the community without a live-in caregiver (family or hired)