To simplify the expression \((9x^2 - 6x + 2) - (x^2 + 4x - 39)\), we first distribute the negative sign through the second polynomial:
\[ (9x^2 - 6x + 2) - (x^2 + 4x - 39) = 9x^2 - 6x + 2 - x^2 - 4x + 39 \]
Now, we combine like terms:
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For the \(x^2\) terms: \[ 9x^2 - x^2 = 8x^2 \]
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For the \(x\) terms: \[ -6x - 4x = -10x \]
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For the constant terms: \[ 2 + 39 = 41 \]
Now, putting it all together, we have:
\[ 8x^2 - 10x + 41 \]
So, the simplified difference of the polynomials is:
\[ \boxed{8x^2 - 10x + 41} \]
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