If both (x + 2) and (2x + 1) are the factors of ax² + 2x + b, prove that a – b = 0.

Let p(x) = ax² + 2x + b

As (x + 2) and (2x + 1) are factors of p(x) hence each will divide p(x) leaving remainder as 0

Let us use the remainder theorem which states that if (x – a) divides a quadratic polynomial p(x) then p(a) = 0

First a = –2

⇒ p(a) = a(–2)² + 2(–2) + b

⇒ 0 = 4a – 4 + b

⇒ b = 4 – 4a …(i)

Now a = –1/2

⇒ p(a) = a(–1/2)² + 2(–1/2) + b

Multiply whole equation by 4

⇒ 0 = a – 4 + 4b

⇒ 4b = 4 – a …(ii)

Subtract (i) from (ii)

⇒ 4b – b = (4 – a) – (4 – 4a)

⇒ 3b = 4 – a – 4 + 4a

⇒ 3b = 3a

⇒ a = b

⇒ a – b = 0

Hence proved