The problem consists of two parts: (a) Simplify the expression $\frac{2x - 3}{x^2 - 9} + \frac{4}{x + 3}$. (b) Solve the equation $5^{2x - 1} = 125$ for $x$.

1. Problem Description

The problem consists of two parts:

(a) Simplify the expression $\frac{2x - 3}{x^2 - 9} + \frac{4}{x + 3}$.

(b) Solve the equation $5^{2x - 1} = 125$ for $x$.

2. Solution Steps

(a) Simplify the expression:

First, factor the denominator $x^2 - 9$ as a difference of squares:

$x^2 - 9 = (x - 3)(x + 3)$

Now rewrite the expression:

$\frac{2x - 3}{(x - 3)(x + 3)} + \frac{4}{x + 3}$

To add the fractions, we need a common denominator, which is $(x - 3)(x + 3)$. Multiply the second fraction by $\frac{x - 3}{x - 3}$:

$\frac{2x - 3}{(x - 3)(x + 3)} + \frac{4(x - 3)}{(x - 3)(x + 3)}$

Combine the fractions:

$\frac{2x - 3 + 4(x - 3)}{(x - 3)(x + 3)}$

Simplify the numerator:

$\frac{2x - 3 + 4x - 12}{(x - 3)(x + 3)}$

$\frac{6x - 15}{(x - 3)(x + 3)}$

Factor out a 3 from the numerator:

$\frac{3(2x - 5)}{(x - 3)(x + 3)}$

(b) Solve the equation:

$5^{2x - 1} = 125$

Since $125 = 5^3$, we can rewrite the equation as:

$5^{2x - 1} = 5^3$

Since the bases are equal, we can equate the exponents:

$2x - 1 = 3$

Add 1 to both sides:

$2x = 4$

Divide both sides by 2:

$x = 2$

3. Final Answer

(a) Simplified expression: $\frac{3(2x - 5)}{(x - 3)(x + 3)}$

(b) Solution for $x$: $x = 2$