To simplify the expression \( \frac{x^2(x-2)(x+5) - x}{x-2} \), let's first simplify the numerator step-by-step.
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Factor out \(x\) (since both terms in the numerator have an \(x\)): \[ x^2(x-2)(x+5) - x = x \left( x(x-2)(x+5) - 1 \right) \]
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Rewrite the expression: \[ = \frac{x \left( x(x-2)(x+5) - 1 \right)}{x-2} \]
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Now let's simplify \( x(x-2)(x+5) - 1 \): \[ x(x-2)(x+5) = x \left( x^2 + 3x - 10 \right) = x^3 + 3x^2 - 10x \] So, the expression becomes: \[ x^3 + 3x^2 - 10x - 1 \]
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The numerator of the rational expression is now: \[ x^3 + 3x^2 - 10x - 1 \]
So, our expression looks like this now: \[ \frac{x(x^3 + 3x^2 - 10x - 1)}{x-2} \]
Since we cannot factor \(x^3 + 3x^2 - 10x - 1\) and it does not share any common factors with \(x-2\), we can write the final expression in a simplified form: \[ \frac{x(x^3 + 3x^2 - 10x - 1)}{x-2} \]
Given that we were instructed to keep the denominator in its factored form, the final simplified rational expression is: \[ \frac{x(x^3 + 3x^2 - 10x - 1)}{x-2} \]
This is the final result.