Question

Express in simplest form: x^3 + 4x^2 − 32x/32x − 8x^2 .
(1 point)
Responses

-x
-x

x3− 18
x3− 18

−(x+8)8
the fraction with numerator negative open paren x plus 8 close paren and denominator 8

x+88

Answer 1

To simplify the expression \(\frac{x^3 + 4x^2 - 32x}{32x - 8x^2}\), we can start by factoring both the numerator and the denominator.

### Step 1: Factor the Numerator
The numerator is:

\[
x^3 + 4x^2 - 32x
\]

We can factor out \(x\):

\[
x(x^2 + 4x - 32)
\]

Next, we can factor the quadratic \(x^2 + 4x - 32\). We need to find two numbers that multiply to \(-32\) and add to \(4\). These numbers are \(8\) and \(-4\):

\[
x^2 + 4x - 32 = (x + 8)(x - 4)
\]

Thus, the numerator becomes:

\[
x(x + 8)(x - 4)
\]

### Step 2: Factor the Denominator
The denominator is:

\[
32x - 8x^2
\]

We can factor out \(8x\):

\[
8x(4 - x)
\]

### Step 3: Rewrite the Expression
Now we can rewrite the expression:

\[
\frac{x(x + 8)(x - 4)}{8x(4 - x)}
\]

### Step 4: Simplify the Expression
We can cancel \(x\) from the numerator and denominator (assuming \(x \neq 0\)):

\[
\frac{(x + 8)(x - 4)}{8(4 - x)}
\]

Notice that \(4 - x\) can be rewritten as \(-1(x - 4)\):

\[
\frac{(x + 8)(x - 4)}{8(-1)(x - 4)} = -\frac{(x + 8)}{8}
\]

Finally, the expression simplifies to:

\[
-\frac{x + 8}{8}
\]

### Conclusion
This is in simplest form, and the correct response is:

**−(x + 8)/8** (the fraction with numerator negative open paren x plus 8 close paren and denominator 8)