To factor the given quadratic expression [tex]\(-3x^2 - 24x - 36\)[/tex], we can follow these steps:
1. Identify the quadratic expression:
[tex]\[
-3x^2 - 24x - 36
\][/tex]
2. Factor out the greatest common factor (GCF):
The GCF of the coefficients [tex]\(-3\)[/tex], [tex]\(-24\)[/tex], and [tex]\(-36\)[/tex] is [tex]\(-3\)[/tex]. We can factor out [tex]\(-3\)[/tex] from the entire expression:
[tex]\[
-3(x^2 + 8x + 12)
\][/tex]
3. Factor the quadratic trinomial:
We now need to factor the trinomial [tex]\(x^2 + 8x + 12\)[/tex]. We look for two numbers that multiply to [tex]\(12\)[/tex] and add to [tex]\(8\)[/tex]. These two numbers are [tex]\(2\)[/tex] and [tex]\(6\)[/tex].
4. Write the trinomial as a product of two binomials:
Using [tex]\(2\)[/tex] and [tex]\(6\)[/tex], the trinomial [tex]\(x^2 + 8x + 12\)[/tex] can be factored as:
[tex]\[
(x + 2)(x + 6)
\][/tex]
5. Combine the factors with the GCF:
Incorporate the GCF back into the factored expression:
[tex]\[
-3(x + 2)(x + 6)
\][/tex]
Therefore, the given expression [tex]\(-3x^2 - 24x - 36\)[/tex] is equivalent to:
[tex]\[
-3 (x + 2) (x + 6)
\][/tex]
