Let's simplify the fraction [tex]\(\frac{-4x^2 + 16}{2x^2 + 2x - 12}\)[/tex] step-by-step.
1. Factor the numerator and the denominator:
- Numerator:
[tex]\( -4x^2 + 16 \)[/tex]
We can factor out [tex]\(-4\)[/tex]:
[tex]\[
-4(x^2 - 4)
\][/tex]
Notice that [tex]\(x^2 - 4\)[/tex] is a difference of squares:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
Therefore, the numerator becomes:
[tex]\[
-4(x - 2)(x + 2)
\][/tex]
- Denominator:
[tex]\( 2x^2 + 2x - 12 \)[/tex]
We can factor out [tex]\(2\)[/tex]:
[tex]\[
2(x^2 + x - 6)
\][/tex]
Notice that [tex]\(x^2 + x - 6\)[/tex] can be factored:
[tex]\[
x^2 + x - 6 = (x + 3)(x - 2)
\][/tex]
Therefore, the denominator becomes:
[tex]\[
2(x + 3)(x - 2)
\][/tex]
2. Rewrite the fraction with the factored forms:
[tex]\[
\frac{-4(x - 2)(x + 2)}{2(x + 3)(x - 2)}
\][/tex]
3. Cancel the common terms:
The term [tex]\((x - 2)\)[/tex] appears in both the numerator and the denominator, so they cancel out:
[tex]\[
\frac{-4(x + 2)}{2(x + 3)}
\][/tex]
4. Simplify the fraction further:
We can simplify the constants:
[tex]\[
\frac{-4(x + 2)}{2(x + 3)} = \frac{-4}{2} \cdot \frac{(x + 2)}{(x + 3)} = -2 \cdot \frac{(x + 2)}{(x + 3)}
\][/tex]
Therefore, the simplified fraction becomes:
[tex]\[
\frac{-2(x + 2)}{x + 3}
\][/tex]
So, the correct answer is [tex]\(\frac{-2(x + 2)}{x + 3}\)[/tex], which matches with option C: [tex]\(\frac{-2x - 4}{x + 3} \)[/tex].
