Solve $x^{2} + x - (a + 2) (a + 1) = 0$ by using quadratic formula.

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Solution

Given $x^{2} + x - (a + 2) (a + 1) = 0$ , Here, a = 1, b = 1, c = -(a + 2)(a + 1)
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- 1 \pm \sqrt{1^{2} - 4 {(a + 2) (a + 1)}}}{2 (1)} = \frac{- 1 \pm \sqrt{1 - 4 {- (a^{2} + 3 a + 2)}}{2}$
$x = \frac{- 1 \pm \sqrt{1 + 4 a^{2} + 12 a + 8}}{2} = \frac{- 1 \pm \sqrt{4 a^{2} + 12 a + 9}}{2} = \frac{- 1 \pm \sqrt{(2 a + 3)^{2}}}{2} = \frac{- 1 \pm (2 a + 3)}{2}$
$∴ x = \frac{- 1 + 2 a + 3}{2}$ and $x = \frac{- 1 - 2 a - 3}{2} ∴ x = \frac{2 a + 2}{2}$ and $x = \frac{- 2 a - 4}{2}$
$x = a + 1 x = - (a + 2)$
$∴$ The roots of $x^{2} + x - (a + 2) (a + 1) = 0$ are $(a + 1)$ and $- (a + 2)$

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