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The expression to simplify is:

\[
\frac{-10x^2 + 20x + 80}{x + 2}
\]

### Step-by-step simplification:

We can simplify this by performing polynomial division. Let's divide \( -10x^2 + 20x + 80 \) by \( x + 2 \).

1. **Divide the first term**: Divide the leading term \( -10x^2 \) by \( x \), which gives \( -10x \).

2. **Multiply back**: Multiply \( -10x \) by \( x + 2 \), which gives \( -10x^2 - 20x \).

3. **Subtract**: Subtract \( (-10x^2 - 20x) \) from \( -10x^2 + 20x \), giving:

\[
(-10x^2 + 20x) - (-10x^2 - 20x) = 40x
\]

4. **Bring down the next term**: Now we have \( 40x + 80 \).

5. **Divide the next term**: Divide \( 40x \) by \( x \), which gives \( 40 \).

6. **Multiply back**: Multiply \( 40 \) by \( x + 2 \), which gives \( 40x + 80 \).

7. **Subtract**: Subtract \( (40x + 80) \) from \( 40x + 80 \), which gives \( 0 \).

So, the quotient is:

\[
-10x + 40
\]

Thus, the simplified form of the expression is \( \boxed{-10x + 40} \).

Would you like more details on any part of this solution?

### Related questions:
1. How do you perform synthetic division with polynomials?
2. What are the steps to factor quadratic expressions?
3. How can you check if a division of polynomials is correct?
4. What are other ways to simplify rational expressions?
5. How do you simplify expressions involving higher-degree polynomials?

### Tip:
Always double-check your division by multiplying the quotient back by the divisor to ensure accuracy!

Which of the following is the simplified form of (-10x^2 + 20x + 80) / (x + 2)?

Understand how to simplify the rational expression (-10x^2 + 20x + 80) / (x + 2) through polynomial division. This step-by-step guide is ideal for students in Grades 9-11, focusing on algebraic simplification techniques.

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