It looks like you have a system of equations that you want to solve. Let's rewrite the system clearly:

1. \(\frac{3}{x+2} - \frac{y}{y+1} = -1\)
2. \(\frac{x}{x+2} + \frac{2}{y+1} = -\frac{5}{3}\)

To solve this system, we can use substitution or elimination methods. However, due to the complexity of the fractions, it might be easier to first simplify the equations.

Let's start with the first equation:

\[
\frac{3}{x+2} - \frac{y}{y+1} = -1
\]

Multiply both sides by \((x+2)(y+1)\) to clear the denominators:

\[
3(y+1) - y(x+2) = -1(x+2)(y+1)
\]

Simplify:

\[
3y + 3 - yx - 2y = -(x+2)(y+1)
\]

Combine like terms:

\[
y(3 - x - 2) + 3 = -(x+2)(y+1)
\]

\[
y(1 - x) + 3 = -(x+2)(y+1)
\]

Now let's look at the second equation:

\[
\frac{x}{x+2} + \frac{2}{y+1} = -\frac{5}{3}
\]

Multiply both sides by \((x+2)(y+1)\) to clear the denominators:

\[
x(y+1) + 2(x+2) = -\frac{5}{3}(x+2)(y+1)
\]

Simplify:

\[
xy + x + 2x + 4 = -\frac{5}{3}(x+2)(y+1)
\]

Combine like terms:

\[
xy + 3x + 4 = -\frac{5}{3}(x+2)(y+1)
\]

At this point, we have two equations with simplified forms. Solving these equations algebraically might still be complex, so let's consider another approach, such as numerical methods or graphing, to find the solution.

However, if you prefer an algebraic solution, you can try isolating one variable in one of the equations and substituting it into the other equation. This might involve more algebraic manipulation and could be quite lengthy.

For now, let's isolate \(y\) in the first simplified equation:

\[
y(1 - x) + 3 = -(x+2)(y+1)
\]

Distribute on the right-hand side:

\[
y(1 - x) + 3 = -xy - x - 2y - 2
\]

Combine like terms:

\[
y(1 - x + xy + 2y) = -x - 5
\]

This equation is still quite complex. To proceed further, you might need to use numerical methods or graphing tools to find the values of \(x\) and \(y\) that satisfy both equations.