Add. \[ \frac{z^{2}-4 z}{z^{2}+4 z+3}+\frac{3 z-2}{z^{2}+4 z+3} \] Simplify. your answer as much as possible

To simplify the expression, first combine the fractions: \[ \frac{z^{2}-4z + (3z-2)}{z^{2}+4z+3} = \frac{z^{2} - z - 2}{z^{2} + 4z + 3} \] Now, simplify the numerator: \[ z^{2} - z - 2 = (z - 2)(z + 1) \] The denominator can also be factored: \[ z^{2} + 4z + 3 = (z + 1)(z + 3) \] Putting it all together gives: \[ \frac{(z - 2)(z + 1)}{(z + 1)(z + 3)} \] Notice that \( (z + 1) \) cancels from the numerator and denominator, assuming \( z \neq -1 \): \[ \frac{z - 2}{z + 3} \] Thus, the simplified expression is: \[ \frac{z - 2}{z + 3} \]