To simplify the expression \(\frac{x^3 + 4x^2 - 32x}{32x - 8x^2}\), we will first factor both the numerator and the denominator.
Step 1: Factor the numerator.
The numerator is \(x^3 + 4x^2 - 32x\). We can factor out an \(x\):
\[ x(x^2 + 4x - 32) \]
Next, we will factor \(x^2 + 4x - 32\). We need two numbers that multiply to \(-32\) and add to \(4\), which are \(8\) and \(-4\):
\[ x^2 + 4x - 32 = (x + 8)(x - 4) \]
So, 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 entire expression.
Now we have:
\[ \frac{x(x + 8)(x - 4)}{-8x(4 - x)} \]
Step 4: Simplify.
Notice that \(4 - x = -(x - 4)\), so we can rewrite the denominator:
\[ -8x(4 - x) = -8x(-1)(x - 4) = 8x(x - 4) \]
Now we can simplify the expression:
\[ \frac{x(x + 8)(x - 4)}{8x(x - 4)} \]
We can cancel \(x\) (assuming \(x \neq 0\)) and \(x - 4\) (assuming \(x \neq 4\)):
\[ \frac{x + 8}{8} \]
Thus, the expression simplifies to:
\[ \frac{x + 8}{8} \]
Conclusion:
The simplest form of the original expression is \(\frac{x + 8}{8}\). This corresponds to the response:
\[ \text{Start Fraction x plus 8 over 8 End Fraction} \]
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