The problem asks us to simplify the expression $\frac{7b}{b^2 - b - 12} - \frac{4b}{b^2 + 2b - 3}$.

1. Problem Description

The problem asks us to simplify the expression $\frac{7b}{b^2 - b - 12} - \frac{4b}{b^2 + 2b - 3}$.

2. Solution Steps

First, we factor the denominators:

$b^2 - b - 12 = (b - 4)(b + 3)$

$b^2 + 2b - 3 = (b + 3)(b - 1)$

Then, we rewrite the expression as:

$\frac{7b}{(b - 4)(b + 3)} - \frac{4b}{(b + 3)(b - 1)}$

The least common denominator is $(b - 4)(b + 3)(b - 1)$.

Now we rewrite each fraction with the common denominator:

$\frac{7b(b - 1)}{(b - 4)(b + 3)(b - 1)} - \frac{4b(b - 4)}{(b - 4)(b + 3)(b - 1)}$

Combine the fractions:

$\frac{7b(b - 1) - 4b(b - 4)}{(b - 4)(b + 3)(b - 1)}$

Expand the numerator:

$\frac{7b^2 - 7b - 4b^2 + 16b}{(b - 4)(b + 3)(b - 1)}$

Simplify the numerator:

$\frac{3b^2 + 9b}{(b - 4)(b + 3)(b - 1)}$

Factor the numerator:

$\frac{3b(b + 3)}{(b - 4)(b + 3)(b - 1)}$

Cancel the common factor $(b + 3)$ from numerator and denominator:

$\frac{3b}{(b - 4)(b - 1)}$

Expand the denominator:

$\frac{3b}{b^2 - b - 4b + 4}$

$\frac{3b}{b^2 - 5b + 4}$

3. Final Answer

$\frac{3b}{b^2 - 5b + 4}$