Question

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\\)
4 great job!
4b use the substitution \\(2x + 2 = p\\) and solve the resulting equation for \\(p\\).
enter your next step here

Explanation:

Step1: Substitute \( p = 2x + 2 \) into the equation

The original equation is \( (2x + 2)^2 + 8(2x + 2) + 12 = 0 \). Replace \( 2x + 2 \) with \( p \), so we get \( p^2 + 8p + 12 = 0 \).

Step2: Solve the quadratic equation for \( p \)

We can factor the quadratic equation \( p^2 + 8p + 12 = 0 \). We need two numbers that multiply to \( 12 \) and add up to \( 8 \). Those numbers are \( 2 \) and \( 6 \). So, the equation factors as \( (p + 2)(p + 6) = 0 \).

Step3: Find the values of \( p \)

Set each factor equal to zero:

• For \( p + 2 = 0 \), we get \( p = -2 \).
• For \( p + 6 = 0 \), we get \( p = -6 \).

Answer:

The solutions for \( p \) are \( p = -2 \) and \( p = -6 \).