Sure, let's simplify the given expression step by step:
Given the expression:
[tex]\[
\frac{-2 x^2 - 2 x}{8 x^2 - 24 x - 32}
\][/tex]
1. Factor the numerator: The numerator is [tex]\(-2 x^2 - 2 x\)[/tex]. We can factor out the common factor of [tex]\(-2x\)[/tex]:
[tex]\[
-2 x^2 - 2 x = -2x(x + 1)
\][/tex]
So the numerator becomes:
[tex]\[
-2x(x + 1)
\][/tex]
2. Factor the denominator: The denominator is [tex]\(8 x^2 - 24 x - 32\)[/tex]. First, we can factor out the greatest common factor which is [tex]\(8\)[/tex]:
[tex]\[
8 x^2 - 24 x - 32 = 8(x^2 - 3x - 4)
\][/tex]
Next, we factor the quadratic expression [tex]\(x^2 - 3x - 4\)[/tex]. We need to find two numbers that multiply to [tex]\(-4\)[/tex] (the constant term) and add to [tex]\(-3\)[/tex] (the coefficient of [tex]\(x\)[/tex]). These numbers are [tex]\(-4\)[/tex] and [tex]\(1\)[/tex]:
[tex]\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\][/tex]
So the denominator becomes:
[tex]\[
8(x - 4)(x + 1)
\][/tex]
3. Write the simplified fraction:
[tex]\[
\frac{-2x(x + 1)}{8(x - 4)(x + 1)}
\][/tex]
4. Cancel out common factors in the numerator and the denominator: Both the numerator and the denominator have a common [tex]\((x + 1)\)[/tex]:
[tex]\[
\frac{-2x \cancel{(x + 1)}}{8(x - 4) \cancel{(x + 1)}} = \frac{-2x}{8(x - 4)}
\][/tex]
5. Simplify the remaining fraction:
[tex]\[
\frac{-2x}{8(x - 4)} = \frac{-2x}{8x - 32}
\][/tex]
By dividing both the numerator and the denominator by [tex]\(2\)[/tex]:
[tex]\[
\frac{-2x}{8(x - 4)} = \frac{-x}{4(x - 4)}
\][/tex]
Thus,
[tex]\[
\frac{-2 x^2-2 x}{8 x^2-24 x-32} = \frac{-x}{4x - 16}
\][/tex]
The correct answer, therefore, is [tex]\(\boxed{\frac{-x}{4 x - 16}}\)[/tex].
