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Consider this expression:
[tex]\[ -3x^2 - 24x - 36 \][/tex]

What expression is equivalent to the given expression?
[tex]\[\square \left( \square \right)\][/tex]



Answer :

Certainly! Let's work on factoring the given quadratic expression step-by-step:

The given expression is:
[tex]$ -3 x^2 - 24 x - 36 $[/tex]

To factor this expression, we first look for a common factor in each term. In this case, we see that each term is divisible by -3.

1. Factor out -3 from each term:
[tex]$ -3 (x^2 + 8x + 12) $[/tex]

Next, we focus on factoring the quadratic expression inside the parentheses:
[tex]$ x^2 + 8x + 12 $[/tex]

We need to find two numbers that multiply to 12 (the constant term) and add up to 8 (the coefficient of [tex]\(x\)[/tex]).

2. Find the pair of numbers:
The numbers 2 and 6 satisfy these conditions because:

[tex]$ 2 \times 6 = 12 $[/tex]
[tex]$ 2 + 6 = 8 $[/tex]

3. Rewrite and factor the quadratic expression:
[tex]$ x^2 + 8x + 12 = (x + 2)(x + 6) $[/tex]

4. Combine it with the factor we originally took out:
[tex]$ -3 (x + 2)(x + 6) $[/tex]

So, the factored form of the original expression [tex]\( -3 x^2 - 24 x - 36 \)[/tex] is:
[tex]$ -3(x + 2)(x + 6) $[/tex]

Hence, the equivalent expression is:
[tex]$ -3(x + 2)(x + 6) $[/tex]