Question

1. consider the equation $(2x + 2)^2 + 8(2x + 2) + 12 = 0$.

4a what substitution can we make to reduce this equation to a monic quadratic in the variable $p$?
$p = 2x + 2$
4b use the substitution $2x + 2 = p$ and solve the resulting equation for $p$.
$p = -2, -6$
4c use your answer to the previous part to find $x$.

Explanation:

Step1: Use \( p = -2 \) to solve for \( x \)

Substitute \( p = 2x + 2 = -2 \) into the equation. Then we have \( 2x + 2 = -2 \). Subtract 2 from both sides: \( 2x = -2 - 2 = -4 \). Divide both sides by 2: \( x = \frac{-4}{2} = -2 \).

Step2: Use \( p = -6 \) to solve for \( x \)

Substitute \( p = 2x + 2 = -6 \) into the equation. Then we have \( 2x + 2 = -6 \). Subtract 2 from both sides: \( 2x = -6 - 2 = -8 \). Divide both sides by 2: \( x = \frac{-8}{2} = -4 \).

Answer:

\( x = -2 \) or \( x = -4 \)