Given that the roots of the equation $\displaystyle x^2 + ax + (a + 2) = 0 $ differ by 2, find the possible values of the constant a. Hence, find the possible values of the roots of the equation.

This is my solution:

$\displaystyle 2\alpha + 2 = -a$____(1)

$\displaystyle \alpha(\alpha + 2) = a + 2$____(2)

From (1),

$\displaystyle \alpha = \frac{-a - 2}{2}$____(3)

Sub (3) into (2),

$\displaystyle (\frac{-a - 2}{2})(\frac{-a - 2}{2} + 2) = a + 2$

$\displaystyle (-a - 2)^2 + 2(-a - 2) - 8 = 0$

$\displaystyle a^2 + 4a + 4 - 2a - 4 -8 = 0$

$\displaystyle a^2 + 2a - 8 = 0$

$\displaystyle (a + 4)(a - 2) = 0$

Hence, I got the answer a = -4 or a = 2 but in the book the answers are a = 6 or a = -2. Where have I gone wrong?