https://file.aiquickdraw.com/mathbot/file/1729619794171gfcoutln.jpg

The given expression to simplify is:

\[
\frac{x^3 + 4x^2 - 32x}{32x - 8x^2}
\]

### Step 1: Factor both the numerator and the denominator.

#### Numerator:
The numerator is \( x^3 + 4x^2 - 32x \). We can factor out \( x \):

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

Now, we need to check if \( x^2 + 4x - 32 \) can be factored further. Using the quadratic formula or factoring by grouping, we find that it doesn't factor nicely, so we leave it as is for now:

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

#### Denominator:
The denominator is \( 32x - 8x^2 \). We can factor out \( -8x \):

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

### Step 2: Simplify the expression.
Now, the expression becomes:

\[
\frac{x(x^2 + 4x - 32)}{-8x(x - 4)}
\]

Since \( x \) appears in both the numerator and the denominator, we can cancel it out (assuming \( x \neq 0 \)):

\[
\frac{x^2 + 4x - 32}{-8(x - 4)}
\]

### Step 3: Interpret the final result.
We now have:

\[
\frac{x^2 + 4x - 32}{-8(x - 4)}
\]

This matches the fourth option in the multiple-choice list:

\[
\frac{x^3 - 1}{8}
\]

However, this seems like a re-arrangement mistake. The correct form should not match the options here

Express in simplest form: (x^3 + 4x^2 - 32x) / (32x - 8x^2).

Learn how to simplify the rational expression (x^3 + 4x^2 - 32x) / (32x - 8x^2) by factoring polynomials and canceling common factors. This algebra problem is suitable for students in Grades 9-10 and covers key algebraic concepts like factoring and simplifying expressions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation