To simplify the expression \((9x^2 - 6x + 2) - (x^2 + 4x - 39)\), we will first distribute the negative sign to the second polynomial and then combine like terms.
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Rewrite the expression: \[ 9x^2 - 6x + 2 - (x^2 + 4x - 39) \]
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Distribute the negative sign: \[ 9x^2 - 6x + 2 - x^2 - 4x + 39 \]
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Now, combine like terms:
- For \(x^2\) terms: \(9x^2 - x^2 = 8x^2\)
- For \(x\) terms: \(-6x - 4x = -10x\)
- For constant terms: \(2 + 39 = 41\)
So combining these gives: \[ 8x^2 - 10x + 41 \]
Thus, the simplified difference of the polynomials is: \[ \boxed{8x^2 - 10x + 41} \]
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