To simplify the expression \( \frac{x^2(x-2)(x+5) - x}{x-2} \), we'll first rewrite the numerator and then factor as needed.
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Start with the expression:
\[ \frac{x^2(x-2)(x+5) - x}{x-2} \]
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Factor out \( x \) from the numerator:
\[ = \frac{x^2(x-2)(x+5) - x(1)}{x-2} \]
Now we have:
\[ = \frac{x \left( x(x-2)(x+5) - 1 \right)}{x-2} \]
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Next, to simplify \( x(x-2)(x+5) - 1 \):
We expand \( x(x-2)(x+5) \):
\[ = x[(x-2)(x+5)] = x[x^2 + 5x - 2x - 10] = x[x^2 + 3x - 10] \]
Putting that back in our expression:
\[ = \frac{x \left( x(x^2 + 3x - 10) - 1 \right)}{x-2} \]
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Expand the numerator:
\[ = x^3 + 3x^2 - 10x - 1 \]
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We cannot factor this simpler form directly. Hence, our expression remains:
\[ \frac{x^3 + 3x^2 - 10x - 1}{x-2} \]
Finally, since further factoring might be complex, the final result in the factored form for the denominator is:
\[ \frac{x^2(x-2)(x+5) - x}{x-2} = \frac{x^3 + 3x^2 - 10x - 1}{x-2} \]
If you wish to divide or use polynomial long division to simplify even further, that would be the next step, yet the main task appears to be maintaining the factored form of the denominator. Thus the result in the factored form is complete with \( x-2 \) in the denominator.