To simplify the given expression \(- (5x^3 + 7x - 8) + (3x - 5)(-x^2 + 7x + 2)\), we will first simplify each part separately and then combine the results.
Step 1: Distributing the negative sign
Starting with the first part:
\[
- (5x^3 + 7x - 8) = -5x^3 - 7x + 8 \]
Step 2: Simplifying the second part
Next, we simplify the second part \((3x - 5)(-x^2 + 7x + 2)\) by using the distributive property (also known as the FOIL method for binomials):
\[ (3x)(-x^2) + (3x)(7x) + (3x)(2) + (-5)(-x^2) + (-5)(7x) + (-5)(2) \]
Calculating each term:
- \(3x \cdot -x^2 = -3x^3\)
- \(3x \cdot 7x = 21x^2\)
- \(3x \cdot 2 = 6x\)
- \(-5 \cdot -x^2 = 5x^2\)
- \(-5 \cdot 7x = -35x\)
- \(-5 \cdot 2 = -10\)
Now, we combine these results:
\[ -3x^3 + 21x^2 + 6x + 5x^2 - 35x - 10 \]
Combining like terms:
- The \(x^3\) term: \(-3x^3\)
- The \(x^2\) terms: \(21x^2 + 5x^2 = 26x^2\)
- The \(x\) terms: \(6x - 35x = -29x\)
- The constant term: \(-10\)
Thus, we have:
\[ -3x^3 + 26x^2 - 29x - 10 \]
Step 3: Combining both parts
Now we combine the results of both steps:
\[ -5x^3 - 7x + 8 + (-3x^3 + 26x^2 - 29x - 10) \]
Combining like terms:
- The \(x^3\) terms: \(-5x^3 - 3x^3 = -8x^3\)
- The \(x^2\) term: \(26x^2\)
- The \(x\) terms: \(-7x - 29x = -36x\)
- The constant terms: \(8 - 10 = -2\)
Final expression
Combining all of these together gives us the final simplified expression:
\[ -8x^3 + 26x^2 - 36x - 2 \]
Thus, the complete simplification of the expression is:
\[ \boxed{-8x^3 + 26x^2 - 36x - 2} \]