To simplify the complex fraction
\[ \frac{\frac{1}{y - 2} + \frac{2}{y + 4}}{\frac{2}{y + 4} - \frac{1}{y - 5}}, \]
we start by simplifying the numerator and the denominator separately.
Step 1: Simplifying the Numerator
The numerator is
\[ \frac{1}{y - 2} + \frac{2}{y + 4}. \]
To combine these fractions, we need a common denominator, which is \((y - 2)(y + 4)\). Rewriting each fraction:
\[ \frac{1}{y - 2} = \frac{1(y + 4)}{(y - 2)(y + 4)} = \frac{y + 4}{(y - 2)(y + 4)} \]
\[ \frac{2}{y + 4} = \frac{2(y - 2)}{(y + 4)(y - 2)} = \frac{2y - 4}{(y - 2)(y + 4)}. \]
Now, adding these together:
\[ \frac{y + 4 + (2y - 4)}{(y - 2)(y + 4)} = \frac{3y}{(y - 2)(y + 4)}. \]
Step 2: Simplifying the Denominator
The denominator is
\[ \frac{2}{y + 4} - \frac{1}{y - 5}. \]
The common denominator for these two fractions is \((y + 4)(y - 5)\). Rewriting each fraction:
\[ \frac{2}{y + 4} = \frac{2(y - 5)}{(y + 4)(y - 5)} = \frac{2y - 10}{(y + 4)(y - 5)}, \]
\[ \frac{1}{y - 5} = \frac{1(y + 4)}{(y - 5)(y + 4)} = \frac{y + 4}{(y - 5)(y + 4)}. \]
Now, subtracting these:
\[ \frac{(2y - 10) - (y + 4)}{(y + 4)(y - 5)} = \frac{2y - 10 - y - 4}{(y + 4)(y - 5)} = \frac{y - 14}{(y + 4)(y - 5)}. \]
Step 3: Putting it All Together
Now we substitute back into our original complex fraction:
\[ \frac{\frac{3y}{(y - 2)(y + 4)}}{\frac{y - 14}{(y + 4)(y - 5)}}. \]
This simplifies to:
\[ \frac{3y}{(y - 2)(y + 4)} \times \frac{(y + 4)(y - 5)}{y - 14} = \frac{3y(y - 5)}{(y - 2)(y - 14)}. \]
Final Result
Thus, the simplified form of the complex fraction is
\[ \frac{3y(y - 5)}{(y - 2)(y - 14)}. \]