To find the expression equivalent to \(\frac{3x^2 + 2x - 5x - 2x^2 + 2}{(x-2)(x-1)(x+2)(x-2)(x+1)(x+2)}\), we will first simplify the numerator.
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Simplify the Numerator: \[ 3x^2 + 2x - 5x - 2x^2 + 2 \] Combine like terms: \[ (3x^2 - 2x^2) + (2x - 5x) + 2 = x^2 - 3x + 2 \]
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Factor the Numerator: The expression \(x^2 - 3x + 2\) can be factored as follows: \[ x^2 - 3x + 2 = (x-1)(x-2) \]
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Identify and Simplify the Denominator: The denominator is: \[ (x-2)(x-1)(x+2)(x-2)(x+1)(x+2) \] Notice that \((x-2)\) and \((x+2)\) repeat, so we can rewrite the denominator as: \[ (x-1)(x-2)^2 (x+2)^2 (x+1) \]
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Write the Complete Expression: Now we put the numerator and denominator together: \[ \frac{(x-1)(x-2)}{(x-1)(x-2)^2 (x+2)^2 (x+1)} \]
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Cancel Common Factors: We can cancel \((x-1)\) and \((x-2)\) from the numerator and denominator: \[ \frac{1}{(x-2)(x+2)^2 (x+1)} \]
Thus, the simplified expression is: \[ \frac{1}{(x-2)(x+2)^2 (x+1)} \]