To simplify the expression $\frac{7}{x-3} + \frac{2x+5}{x+8}$, we need to combine the two fractions. To do so, we find a common denominator.

Steps:

1. Find the common denominator: The common denominator between $x-3$ and $x+8$ is $(x-3)(x+8)$.

2. Rewrite both fractions with the common denominator: $\frac{7}{x-3} = \frac{7(x+8)}{(x-3)(x+8)}$ $\frac{2x+5}{x+8} = \frac{(2x+5)(x-3)}{(x-3)(x+8)}$

3. Combine the fractions: Now that both fractions have the same denominator, we can combine them: $\frac{7(x+8) + (2x+5)(x-3)}{(x-3)(x+8)}$

4. Expand the numerators: First, expand the terms: $7(x+8) = 7x + 56$ $(2x+5)(x-3) = 2x(x-3) + 5(x-3) = 2x^2 - 6x + 5x - 15 = 2x^2 - x - 15$

   So, the numerator becomes: $7x + 56 + 2x^2 - x - 15 = 2x^2 + 6x + 41$

5. Final expression: The simplified expression is: $\frac{2x^2 + 6x + 41}{(x-3)(x+8)}$

Result:

The simplified form of $\frac{7}{x-3} + \frac{2x+5}{x+8}$ is: $\frac{2x^2 + 6x + 41}{(x-3)(x+8)}$

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Related Questions:

1. How can I factor polynomials like $2x^2 + 6x + 41$?
2. What happens if $x = 3$ or $x = -8$ in this expression?
3. Can this expression be simplified further if additional conditions are given?
4. How do you perform partial fraction decomposition on this type of expression?
5. What are the general steps for finding a common denominator with algebraic expressions?

Tip: Always check if the numerator can be factored further after combining fractions; this helps simplify the expression further.