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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:
1. For the \(x^2\) terms:
\[
9x^2 - x^2 = 8x^2
\]
2. For the \(x\) terms:
\[
-6x - 4x = -10x
\]
3. 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:
\[
\boxed{8x^2 - 10x + 41}
\]