$\frac{x}{x^{2}+3 x+2}-\frac{3 x-2}{x^{2}+8 x+7}$

Step 1 ：Factor the denominators: $\frac{x}{(x+1)(x+2)} - \frac{3x-2}{(x+1)(x+7)}$

Step 2 ：Find a common denominator: $(x+1)(x+2)(x+7)$

Step 3 ：Combine the fractions: $\frac{-2x^2 + 3x + 4}{(x+1)(x+2)(x+7)}$

Step 4 ：Expand the denominator: $\frac{-2x^2 + 3x + 4}{x^3 + 10x^2 + 23x + 14}$

Step 5 ：\(\boxed{\frac{-2x^2 + 3x + 4}{x^3 + 10x^2 + 23x + 14}}\)