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The problem you've shared is:

\[
\frac{x^2}{x - 8} + \frac{7x + 8}{8 - x}
\]

### Step 1: Simplify the second fraction

Notice that \( 8 - x \) is the negative of \( x - 8 \). In other words:

\[
8 - x = -(x - 8)
\]

Thus, we can rewrite the second fraction as:

\[
\frac{7x + 8}{8 - x} = -\frac{7x + 8}{x - 8}
\]

### Step 2: Combine the fractions

Now the expression becomes:

\[
\frac{x^2}{x - 8} - \frac{7x + 8}{x - 8}
\]

Since both fractions have the same denominator \( (x - 8) \), we can combine them:

\[
\frac{x^2 - (7x + 8)}{x - 8}
\]

### Step 3: Simplify the numerator

Distribute the negative sign in the numerator:

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

So the expression becomes:

\[
\frac{x^2 - 7x - 8}{x - 8}
\]

### Step 4: Factor the numerator

Now, factor \( x^2 - 7x - 8 \). The factors of \(-8\) that add up to \(-7\) are \(-8\) and \(1\). Thus:

\[
x^2 - 7x - 8 = (x - 8)(x + 1)
\]

### Step 5: Cancel common factors

Now the expression becomes:

\[
\frac{(x - 8)(x + 1)}{x - 8}
\]

We can cancel out the \(x - 8\) terms (assuming \(x \neq 8\)):

\[
x + 1
\]

### Final Answer:
The simplified expression is:

\[
x + 1 \quad \text{for} \quad x \neq 8
\]

Would you like more details on any of the steps? Here are 5 related questions for further exploration:

1. What happens if \( x = 8 \)? Why can't we simplify the expression for that value?
2. How do we factor trinomials like \( x^2 - 7x - 8 \)?
3. What is the significance of canceling terms in a rational expression?
4. How can we check the original and simplified expressions for equivalent values?
5. Are there any restrictions on rational functions that come from the denominator?

**Tip**: Always check for restrictions (values that make the denominator zero) when simplifying rational expressions.

Solve the expression \( \frac{x^2}{x - 8} + \frac{7x + 8}{8 - x} \)

Learn how to simplify the rational expression \( \frac{x^2}{x - 8} + \frac{7x + 8}{8 - x} \) by factoring and canceling common terms. This problem is ideal for high school students in Grades 10-12 and covers key algebraic techniques in rational expression simplification.

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