To factor the quadratic expression [tex]\(-3x^2 - 24x - 36\)[/tex], follow these steps:
1. Factor out the common term:
The entire expression has a common factor of [tex]\(-3\)[/tex]. So, we can factor it out:
[tex]\[
-3(x^2 + 8x + 12)
\][/tex]
2. Factor the quadratic expression inside the parentheses:
We need to factor [tex]\(x^2 + 8x + 12\)[/tex].
- Identify two numbers that multiply to 12 (the constant term) and add up to 8 (the coefficient of the [tex]\(x\)[/tex] term).
- The numbers 2 and 6 satisfy this condition because [tex]\(2 \times 6 = 12\)[/tex] and [tex]\(2 + 6 = 8\)[/tex].
3. Write the factored form of the quadratic expression:
Therefore, the quadratic expression [tex]\(x^2 + 8x + 12\)[/tex] can be factored as:
[tex]\[
(x + 6)(x + 2)
\][/tex]
4. Include the common factor:
Now, reintroduce the common factor [tex]\(-3\)[/tex]:
[tex]\[
-3(x + 6)(x + 2)
\][/tex]
Therefore, the expression [tex]\(-3x^2 - 24x - 36\)[/tex] is equivalent to [tex]\(-3(x + 6)(x + 2)\)[/tex].
So, the answer is:
[tex]\[
-3 (x + 6) (x + 2)
\][/tex]
