The problem requires solving the rational equation $\frac{24}{5x+2} = \frac{4x+1}{5x^2+12x+4} + \frac{4}{x+2}$.

1. Problem Description

The problem requires solving the rational equation $\frac{24}{5x+2} = \frac{4x+1}{5x^2+12x+4} + \frac{4}{x+2}$.

2. Solution Steps

First, factor the quadratic expression in the denominator: $5x^2+12x+4 = (5x+2)(x+2)$.

The equation becomes:

$\frac{24}{5x+2} = \frac{4x+1}{(5x+2)(x+2)} + \frac{4}{x+2}$

To eliminate the fractions, multiply both sides of the equation by the least common denominator, which is $(5x+2)(x+2)$:

$(5x+2)(x+2) \cdot \frac{24}{5x+2} = (5x+2)(x+2) \cdot \frac{4x+1}{(5x+2)(x+2)} + (5x+2)(x+2) \cdot \frac{4}{x+2}$

$24(x+2) = (4x+1) + 4(5x+2)$

$24x + 48 = 4x+1 + 20x + 8$

$24x + 48 = 24x + 9$

Subtract $24x$ from both sides:

$48 = 9$

Since $48=9$ is a false statement, there is no solution to the equation.

Also we should note that $5x+2 \neq 0$ so $x \neq -2/5$, and $x+2 \neq 0$ so $x \neq -2$.

3. Final Answer

B. The equation has no solution.