Question

use the zero product property to find the solutions to the equation \\((x + 2)(x + 3) = 12\\)

\\(x = -6\\) or \\(x = 1\\)
\\(x = -3\\) or \\(x = -2\\)
\\(x = -1\\) or \\(x = 6\\)
\\(x = 2\\) or \\(x = 3\\)

Explanation:

Expand the left side of the equation

$$ (x + 2)(x + 3) = 12 $$

$$ x^2 + 5x + 6 = 12 $$

Set the quadratic equation to zero

$$ x^2 + 5x + 6 - 12 = 0 $$

$$ x^2 + 5x - 6 = 0 $$

Factor and apply the zero product property

$$ (x + 6)(x - 1) = 0 $$

$$ x + 6 = 0 \quad \text{or} \quad x - 1 = 0 $$

$$ x = -6 \quad \text{or} \quad x = 1 $$

Answer:

• (A) \(x = -6\) or \(x = 1\) (Correct answer)
• (B) \(x = -3\) or \(x = -2\)
• (C) \(x = -1\) or \(x = 6\)
• (D) \(x = 2\) or \(x = 3\)