To simplify the expression \( (9x^2 - 6x + 2) - (x^2 + 4x - 39) \), we follow these steps:
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Distribute the negative sign across the second polynomial: \[ 9x^2 - 6x + 2 - x^2 - 4x + 39 \]
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Combine like terms:
- Combine the \( x^2 \) terms: \( 9x^2 - x^2 = 8x^2 \)
- Combine the \( x \) terms: \( -6x - 4x = -10x \)
- Combine the constant terms: \( 2 + 39 = 41 \)
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Putting it all together, we have: \[ 8x^2 - 10x + 41 \]
Thus, the simplified difference of the polynomial is \( \boxed{8x^2 - 10x + 41} \).