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]
