Option 1 : 65

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**Given:**

x + y = 4

**Formula used:**

(x + y)^{3} = x^{3} + y^{3} + 3xy(x + y)

**Calculations:**

(x + y)3 = x3 + y3 + 3xy(x + y)

⇒ (4)^{3} = x3 + y3 + 3xy × 4

⇒ (4)3 = x3 + y3 + 12xy

⇒ 64 = x3 + y3 + 12xy

⇒ x3 + y3 + 12xy + 1 = 64 + 1

⇒ x3 + y3 + 12xy + 1 = 65

**∴ The value of x3 + y3 + 12xy + 1 is 65.**

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