The problem is to solve the equation $\frac{11}{x^2+x} + \frac{10}{x} = \frac{7}{x+1}$ for $x$ and check the solution.

1. Problem Description

The problem is to solve the equation $\frac{11}{x^2+x} + \frac{10}{x} = \frac{7}{x+1}$ for $x$ and check the solution.

2. Solution Steps

First, we factor the denominator $x^2 + x$ as $x(x+1)$. Then the equation becomes:

$\frac{11}{x(x+1)} + \frac{10}{x} = \frac{7}{x+1}$

To eliminate the fractions, we multiply both sides of the equation by the least common denominator, which is $x(x+1)$:

$x(x+1) \left( \frac{11}{x(x+1)} + \frac{10}{x} \right) = x(x+1) \left( \frac{7}{x+1} \right)$

Distribute $x(x+1)$ to each term:

$11 + 10(x+1) = 7x$

$11 + 10x + 10 = 7x$

$10x + 21 = 7x$

Subtract $7x$ from both sides:

$10x - 7x + 21 = 0$

$3x + 21 = 0$

Subtract 21 from both sides:

$3x = -21$

Divide both sides by 3:

$x = -7$

Now, we check the solution $x=-7$ in the original equation:

$\frac{11}{(-7)^2 + (-7)} + \frac{10}{-7} = \frac{7}{-7+1}$

$\frac{11}{49 - 7} + \frac{10}{-7} = \frac{7}{-6}$

$\frac{11}{42} - \frac{10}{7} = -\frac{7}{6}$

$\frac{11}{42} - \frac{60}{42} = -\frac{49}{42}$

$\frac{11-60}{42} = -\frac{49}{42}$

$\frac{-49}{42} = -\frac{49}{42}$

Since the left side equals the right side, the solution $x = -7$ is valid.

3. Final Answer

A. $x = -7$