To factor the quadratic expression [tex]\(-3x^2 - 24x - 36\)[/tex], we start by factoring out the common factor from the terms. Here’s the step-by-step breakdown:
1. Identify the greatest common factor (GCF) for the terms in the expression [tex]\(-3x^2 - 24x - 36\)[/tex]. In this case, the GCF is [tex]\(-3\)[/tex].
2. Factor [tex]\(-3\)[/tex] out of the entire expression:
[tex]\[
-3(x^2 + 8x + 12)
\][/tex]
3. Next, focus on factoring the quadratic expression inside the parentheses: [tex]\(x^2 + 8x + 12\)[/tex].
4. Find two numbers that multiply to [tex]\(12\)[/tex] (the constant term) and add up to [tex]\(8\)[/tex] (the coefficient of the linear term, [tex]\(x\)[/tex]). These numbers are [tex]\(2\)[/tex] and [tex]\(6\)[/tex].
5. Express [tex]\(x^2 + 8x + 12\)[/tex] as a product of two binomials:
[tex]\[
(x + 2)(x + 6)
\][/tex]
6. Combine this factored form with the factor of [tex]\(-3\)[/tex] that was originally factored out:
[tex]\[
-3(x + 2)(x + 6)
\][/tex]
Thus, the equivalent expression to [tex]\(-3x^2 - 24x - 36\)[/tex] is [tex]\(-3(x + 2)(x + 6)\)[/tex].
Therefore, the correct expression is:
[tex]$\boxed{-3} \boxed{(x + 2)(x + 6)}$[/tex]
