Simplixy \( \frac{x^{2}+4 x+3}{x^{2}+5 x+4}-\frac{x^{2}-9}{x^{2}+7 x+12} \)

To simplify the expression \[ \frac{x^{2}+4 x+3}{x^{2}+5 x+4}-\frac{x^{2}-9}{x^{2}+7 x+12}, \] we start by factoring each of the quadratic expressions in the denominators and numerators. 1. **Factoring the first fraction:** \[ x^{2}+4x+3 = (x+1)(x+3) \] \[ x^{2}+5x+4 = (x+1)(x+4) \] Therefore, the first fraction becomes: \[ \frac{(x+1)(x+3)}{(x+1)(x+4)} = \frac{x+3}{x+4} \quad \text{(for } x \neq -1\text{)} \] 2. **Factoring the second fraction:** \[ x^{2}-9 = (x-3)(x+3) \] \[ x^{2}+7x+12 = (x+3)(x+4) \] The second fraction simplifies to: \[ \frac{(x-3)(x+3)}{(x+3)(x+4)} = \frac{x-3}{x+4} \quad \text{(for } x \neq -3\text{)} \] 3. **Now substituting back into the original expression:** We have: \[ \frac{x+3}{x+4} - \frac{x-3}{x+4} \] Since both fractions have the same denominator, we can combine them: \[ = \frac{(x+3) - (x-3)}{x+4} = \frac{x + 3 - x + 3}{x+4} = \frac{6}{x+4} \quad \text{(for } x \neq -1 \text{ and } x \neq -4 \text{)} \] Thus, the simplified expression is: \[ \boxed{\frac{6}{x+4}} \]