To simplify the given expression \(\frac{x^3 + 4x^2 - 32x}{32x - 8x^2}\), we will first factor both the numerator and the denominator.
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Factoring the numerator \(x^3 + 4x^2 - 32x\):
- We can factor out \(x\): \[ x(x^2 + 4x - 32) \]
- Now we need to factor \(x^2 + 4x - 32\). We look for two numbers that multiply to \(-32\) and add to \(4\). These numbers are \(8\) and \(-4\): \[ x^2 + 4x - 32 = (x + 8)(x - 4) \]
- Therefore, the numerator becomes: \[ x(x + 8)(x - 4) \]
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Factoring the denominator \(32x - 8x^2\):
- We can factor out \(8x\): \[ 8x(4 - x) \]
- We can rearrange \(4 - x\) as \(-(x - 4)\), giving us: \[ 8x(-1)(x - 4) = -8x(x - 4) \]
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Now we can rewrite the entire expression: \[ \frac{x(x + 8)(x - 4)}{-8x(x - 4)} \]
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Canceling common factors:
- Both the numerator and denominator have \(x - 4\) as a common factor which can be canceled out, assuming \(x \neq 4\): \[ \frac{x(x + 8)}{-8x} \]
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Further simplifying:
- Canceling \(x\) from the numerator and the denominator (assuming \(x \neq 0\)): \[ \frac{x + 8}{-8} \]
- This can be expressed as: \[ -\frac{x + 8}{8} \]
Thus, the expression in simplest form is:
\[ -\frac{x + 8}{8} \]
Among the responses you provided, it seems that the correct answer is: \[ -\frac{(x + 8)}{8} \]
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