To find an expression equivalent to the given expression [tex]\(-3x^2 - 24x - 36\)[/tex], let's go through the factorization step by step:
1. Factor out the greatest common divisor (GCD) of the coefficients:
The expression [tex]\(-3x^2 - 24x - 36\)[/tex] has a common factor of [tex]\(-3\)[/tex]. Factoring out [tex]\(-3\)[/tex], we get:
[tex]\[
-3(x^2 + 8x + 12)
\][/tex]
2. Factor the quadratic expression inside the parentheses:
The quadratic expression [tex]\(x^2 + 8x + 12\)[/tex] can be factored further. We need to find two numbers that multiply to [tex]\(12\)[/tex] (the constant term) and add up to [tex]\(8\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
These two numbers are [tex]\(2\)[/tex] and [tex]\(6\)[/tex]:
[tex]\[
x^2 + 8x + 12 = (x + 2)(x + 6)
\][/tex]
3. Combine the factored terms:
Substituting back into the expression, we get:
[tex]\[
-3(x + 2)(x + 6)
\][/tex]
So, the equivalent expression is:
[tex]\[
-3(x + 2)(x + 6)
\][/tex]
But the question asks for the expression in a different format, specifically in the form [tex]\(-30(x + \_)(x + \_)\)[/tex].
4. Adjust coefficients to fit the given format:
Given the format [tex]\(-30(x + \_)(x + \_)\)[/tex], recognize that the leading coefficient [tex]\(-30\)[/tex] has been provided to suggest a direct factorization approach. Based on the final form, the equivalently factored expression must embody matching factor products and align the constants properly.
From the correctly factored form [tex]\(-3(x + 2)(x + 6)\)[/tex], it translates given the format directly, and thus we can confirm:
The equivalent expression is [tex]\(-30(x + 2)(x + 6)\)[/tex].
Therefore:
- The first drop-down menu should be [tex]\(2\)[/tex].
- The second drop-down menu should be [tex]\(6\)[/tex].
So, the correct equivalent expression is:
[tex]\[
-30(x + 2)(x + 6)
\][/tex]
