Question

Perform the operation(s) and simplify completely. −(5x3+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 across the first polynomial.
2. Expand the second part by using the distributive property.
3. Combine like terms.

Let's start with the first part:

### Step 1: Distributing the negative sign

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

So now our expression looks like:

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

### Step 2: Expanding the second part

We will expand \( (3x - 5)(-x^2 + 7x + 2) \) using the distributive property:

- First, distribute \( 3x \):
\[
3x \cdot (-x^2) + 3x \cdot 7x + 3x \cdot 2 = -3x^3 + 21x^2 + 6x
\]

- Next, distribute \( -5 \):
\[
-5 \cdot (-x^2) + -5 \cdot 7x + -5 \cdot 2 = 5x^2 - 35x - 10
\]

Now combine these two results:

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

Combine like terms:

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

### Step 3: Combine everything together

Now we substitute back into the original expression:

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

This becomes:

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

Combine like terms:

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

This results in:

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

Thus, the final simplified expression is:

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