To simplify the expression \((9x^2 - 6x + 2) - (x^2 + 4x - 39)\), we will first distribute the negative sign across the second polynomial, and then combine like terms.
Start by rewriting the expression:
\[ 9x^2 - 6x + 2 - (x^2 + 4x - 39) \]
Distributing the negative sign gives us:
\[ 9x^2 - 6x + 2 - x^2 - 4x + 39 \]
Now, combine like terms:
- For the \(x^2\) terms: \(9x^2 - x^2 = 8x^2\)
- For the \(x\) terms: \(-6x - 4x = -10x\)
- For the constant terms: \(2 + 39 = 41\)
Putting it all together, we have:
\[ 8x^2 - 10x + 41 \]
Thus, the difference of the polynomials is
\(\boxed{8x^2 - 10x + 41}\).