Simplity Each Be sure to put numeratar and senominatar in parenthesis as needed. \( \frac{z^{2}+5 x+4}{x^{2}-3 x-4} \cdot \frac{-2 x+8}{z^{2}+x-12}=\square \) \( \frac{-3 x+3}{x^{2}+4 x+3} \div \frac{x^{2}+3 x-4}{x^{2}+5 x+4}=\square \)

Let's simplify each expression step by step. ### First Expression \[ \frac{z^{2}+5x+4}{x^{2}-3x-4} \cdot \frac{-2x+8}{z^{2}+x-12} \] 1. **Factor each part:** - For \( z^{2} + 5x + 4 \), it cannot be factored further without specific values for \( z \) and \( x \). - For \( x^{2} - 3x - 4 \): \[ x^{2} - 3x - 4 = (x - 4)(x + 1) \] - For \( -2x + 8 \): \[ -2x + 8 = -2(x - 4) \] - For \( z^{2} + x - 12 \), it cannot be factored further without specific values for \( z \) and \( x \). 2. **Substituting the factors:** \[ \frac{z^{2} + 5x + 4}{(x - 4)(x + 1)} \cdot \frac{-2(x - 4)}{z^{2} + x - 12} \] 3. **Cancel common factors:** The \( (x - 4) \) in the numerator and denominator cancels out: \[ = \frac{(z^{2} + 5x + 4)(-2)}{(x + 1)(z^{2} + x - 12)} \] 4. **Final result for the first expression:** \[ = \frac{-2(z^{2} + 5x + 4)}{(x + 1)(z^{2} + x - 12)} \] ### Second Expression \[ \frac{-3x + 3}{x^{2} + 4x + 3} \div \frac{x^{2} + 3x - 4}{x^{2} + 5x + 4} \] 1. **Rewrite the division as multiplication by the reciprocal:** \[ = \frac{-3x + 3}{x^{2} + 4x + 3} \cdot \frac{x^{2} + 5x + 4}{x^{2} + 3x - 4} \] 2. **Factor each part:** - For \( -3x + 3 \): \[ -3x + 3 = -3(x - 1) \] - For \( x^{2} + 4x + 3 \): \[ x^{2} + 4x + 3 = (x + 3)(x + 1) \] - For \( x^{2} + 5x + 4 \): \[ x^{2} + 5x + 4 = (x + 4)(x + 1) \] - For \( x^{2} + 3x - 4 \): \[ x^{2} + 3x - 4 = (x + 4)(x - 1) \] 3. **Substituting the factors:** \[ = \frac{-3(x - 1)}{(x + 3)(x + 1)} \cdot \frac{(x + 4)(x + 1)}{(x + 4)(x - 1)} \] 4. **Cancel common factors:** The \( (x + 4) \) cancels out, and \( (x - 1) \) cancels out: \[ = \frac{-3}{(x + 3)(x + 1)} \] 5. **Final result for the second expression:** \[ = \frac{-3}{(x + 3)(x + 1)} \] ### Summary of Results 1. First expression: \[ \frac{-2(z^{2} + 5x + 4)}{(x + 1)(z^{2} + x - 12)} \] 2. Second expression: \[ \frac{-3}{(x + 3)(x + 1)} \]