To determine what expression is equivalent to [tex]\(-3x^2 - 24x - 36\)[/tex], we need to factorize the quadratic expression. Here is the step-by-step solution:
1. Identify the common factor:
The first step in factorizing a quadratic expression is to determine if there is a common factor for all terms. In this case, [tex]\(-3\)[/tex] is a common factor.
2. Factor out the common factor:
We can factor [tex]\(-3\)[/tex] out of each term in the expression:
[tex]\[
-3(x^2 + 8x + 12)
\][/tex]
3. Factorize the quadratic expression inside the parentheses:
Now, we need to factorize [tex]\(x^2 + 8x + 12\)[/tex].
- Look for two numbers that multiply to [tex]\(12\)[/tex] (the constant term) and add up to [tex]\(8\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
- Those numbers are [tex]\(6\)[/tex] and [tex]\(2\)[/tex] because [tex]\(6 \times 12 = 12\)[/tex] and [tex]\(6 + 2 = 8\)[/tex].
4. Write the quadratic expression as a product of two binomials:
Therefore, [tex]\(x^2 + 8x + 12\)[/tex] can be written as:
[tex]\[
(x + 6)(x + 2)
\][/tex]
5. Combine the factored terms:
Substituting back into the original expression, we get:
[tex]\[
-3(x + 6)(x + 2)
\][/tex]
Hence, the factored form of the given expression [tex]\(-3x^2 - 24x - 36\)[/tex] is [tex]\(-3(x + 6)(x + 2)\)[/tex].
