Perform the operation(s) and simplify completely. −(5x^3+7x−8)+(3x−5)(−x2+7x+2)

Answer 1

To simplify the expression \(- (5x^3 + 7x - 8) + (3x - 5)(-x^2 + 7x + 2)\), we will follow these steps:

1. Distribute the negative sign in the first term.
2. Multiply the second term using the distributive property (FOIL).
3. Combine like terms.

Step 1: Distribute the negative sign

\[ -(5x^3 + 7x - 8) = -5x^3 - 7x + 8 \]

Step 2: Expand the second term

Now we expand \((3x - 5)(-x^2 + 7x + 2)\):

Using the distributive property:

\[ (3x)(-x^2) + (3x)(7x) + (3x)(2) + (-5)(-x^2) + (-5)(7x) + (-5)(2) \]

Calculating each term:

\[ = -3x^3 + 21x^2 + 6x + 5x^2 - 35x - 10 \]

Combine the like terms from this expansion:

\[ = -3x^3 + (21x^2 + 5x^2) + (6x - 35x) - 10 \] \[ = -3x^3 + 26x^2 - 29x - 10 \]

Step 3: Combine both parts of the expression

Now we combine the results from Step 1 and Step 2:

\[ -5x^3 - 7x + 8 + (-3x^3 + 26x^2 - 29x - 10) \]

Combining like terms:

• For \(x^3\): \(-5x^3 - 3x^3 = -8x^3\)
• For \(x^2\): \(0 + 26x^2 = 26x^2\)
• For \(x\): \(-7x - 29x = -36x\)
• For the constant term: \(8 - 10 = -2\)

Putting it all together, we get:

\[ -8x^3 + 26x^2 - 36x - 2 \]

Thus, the final simplified expression is:

\[ \boxed{-8x^3 + 26x^2 - 36x - 2} \]