Answer

We have, f(x) = x² + px + 45

Let zeroes be α and β

We know that, if α and β are the roots of a quadratic polynomial ax² + bx + c then

and

Here, α + β = -p and αβ = 45

Given,

(α – β)² = 144

⇒ α² + β² – 2αβ = 144

⇒ (α + β)² – 2αβ – 2αβ = 144

⇒ (α + β)² – 4αβ = 144

⇒ (-p)² – 4(45) = 144

⇒ p² – 180 = 144

⇒ p² = 324

⇒ p = ±18