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The expression given is:

\[
\frac{x^2 - 7}{x + 5} + \frac{x + 3}{8}
\]

### Step 1: Find a Common Denominator
To add these two fractions, we need a common denominator. The denominators are \(x + 5\) and \(8\). The common denominator will be the product of these two:

\[
\text{Common Denominator} = 8(x + 5)
\]

### Step 2: Rewrite Each Fraction with the Common Denominator
Rewrite each fraction so that they both have this common denominator:

1. The first fraction \(\frac{x^2 - 7}{x + 5}\) becomes:

\[
\frac{8(x^2 - 7)}{8(x + 5)}
\]

2. The second fraction \(\frac{x + 3}{8}\) becomes:

\[
\frac{(x + 3)(x + 5)}{8(x + 5)}
\]

### Step 3: Combine the Fractions
Now that the fractions have the same denominator, we can combine them:

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

### Step 4: Expand and Simplify the Numerator
Expand the expressions in the numerator:

- \(8(x^2 - 7) = 8x^2 - 56\)
- \((x + 3)(x + 5)\) expands as \(x^2 + 5x + 3x + 15 = x^2 + 8x + 15\)

Now, add these together:

\[
8x^2 - 56 + x^2 + 8x + 15
\]

Combine like terms:

\[
(8x^2 + x^2) + 8x + (-56 + 15) = 9x^2 + 8x - 41
\]

So, the expression becomes:

\[
\frac{9x^2 + 8x - 41}{8(x + 5)}
\]

### Final Answer
The simplified expression is:

\[
\frac{9x^2 + 8x - 41}{8(x + 5)}
\]

Would you like more details or have any questions?

Here are five related questions to explore further:

1. How do you determine the least common denominator when adding fractions with different denominators?
2. What are the steps to factorize a quadratic expression like \(9x^2 + 8x - 41\)?
3. How can you simplify complex rational expressions involving multiple terms?
4. What happens if the denominator includes a binomial squared, such as \((x + 5)^2\)?
5. How do you verify that the simplified expression is equivalent to the original one?

**Tip:** When simplifying fractions, always expand and combine like terms carefully to avoid mistakes in the final simplified expression.

Learn how to add fractions with different denominators using a common denominator. Step-by-step solution for the expression (x^2 - 7)/(x + 5) + (x + 3)/8, covering key algebraic concepts like fractions and algebraic manipulation. Suitable for Grades 9-12 students and anyone interested in algebraic fraction addition.

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