Find the difference: \( \begin{array}{l}\frac{2}{x+10}-\frac{3}{x+4} \\ \frac{-x+38}{(x+10)(x+4)} \\ -\frac{-x-22}{(x+10)(x+4)} \\ -\frac{1}{6}\end{array} \)

To find the difference, we will simplify each expression step by step: 1. The first part is \( \frac{2}{x+10} - \frac{3}{x+4} \). To combine these fractions, we will find a common denominator, which is \((x+10)(x+4)\). \[ \frac{2}{x+10} = \frac{2(x+4)}{(x+10)(x+4)} = \frac{2x + 8}{(x+10)(x+4)} \] \[ \frac{3}{x+4} = \frac{3(x+10)}{(x+10)(x+4)} = \frac{3x + 30}{(x+10)(x+4)} \] Now we can combine the two fractions: \[ \frac{2x + 8}{(x+10)(x+4)} - \frac{3x + 30}{(x+10)(x+4)} = \frac{(2x + 8) - (3x + 30)}{(x+10)(x+4)} = \frac{2x + 8 - 3x - 30}{(x+10)(x+4)} \] This simplifies to: \[ \frac{-x - 22}{(x+10)(x+4)} \] 2. The second part is \( -\frac{-x-22}{(x+10)(x+4)} \). Simplifying this gives: \[ \frac{x + 22}{(x+10)(x+4)} \] 3. Now we need to combine all of this with \( -\frac{1}{6} \): \[ \frac{-x - 22}{(x+10)(x+4)} + \frac{x + 22}{(x+10)(x+4)} - \frac{1}{6} \] Adding the first two fractions results in: \[ \frac{(-x - 22) + (x + 22)}{(x+10)(x+4)} = \frac{0}{(x+10)(x+4)} = 0 \] 4. Therefore, we have: \[ 0 - \frac{1}{6} = -\frac{1}{6} \] In conclusion, the final difference is: \[ \boxed{-\frac{1}{6}} \]