19. \( \frac{8}{m^{2}+6 m+5}+\frac{4}{m+1} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{8}{m^{2}+6m+5}+\frac{4}{m+1}\) - step1: Factor the expression: \(\frac{8}{\left(m+5\right)\left(m+1\right)}+\frac{4}{m+1}\) - step2: Reduce fractions to a common denominator: \(\frac{8}{\left(m+5\right)\left(m+1\right)}+\frac{4\left(m+5\right)}{\left(m+1\right)\left(m+5\right)}\) - step3: Rewrite the expression: \(\frac{8}{\left(m+5\right)\left(m+1\right)}+\frac{4\left(m+5\right)}{\left(m+5\right)\left(m+1\right)}\) - step4: Transform the expression: \(\frac{8+4\left(m+5\right)}{\left(m+5\right)\left(m+1\right)}\) - step5: Multiply the terms: \(\frac{8+4m+20}{\left(m+5\right)\left(m+1\right)}\) - step6: Add the numbers: \(\frac{28+4m}{\left(m+5\right)\left(m+1\right)}\) - step7: Multiply the terms: \(\frac{28+4m}{m^{2}+6m+5}\) To simplify the expression \( \frac{8}{m^{2}+6 m+5}+\frac{4}{m+1} \), we follow these steps: 1. **Identify the denominators**: The first term has a denominator of \( m^{2}+6m+5 \) and the second term has a denominator of \( m+1 \). 2. **Factor the first denominator**: The quadratic \( m^{2}+6m+5 \) can be factored as \( (m+1)(m+5) \). 3. **Rewrite the expression**: The expression can now be rewritten as: \[ \frac{8}{(m+1)(m+5)} + \frac{4}{m+1} \] 4. **Find a common denominator**: The common denominator for both fractions is \( (m+1)(m+5) \). 5. **Rewrite the second term**: The second term can be rewritten to have the common denominator: \[ \frac{4(m+5)}{(m+1)(m+5)} \] 6. **Combine the fractions**: \[ \frac{8 + 4(m+5)}{(m+1)(m+5)} = \frac{8 + 4m + 20}{(m+1)(m+5)} = \frac{4m + 28}{(m+1)(m+5)} \] Thus, the simplified expression is: \[ \frac{4m + 28}{m^{2} + 6m + 5} \]