Simplify $ \dfrac{y^2-4y-5}{y^2+5y+4} $ to $ \dfrac{y-5}{y+4} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{y+1}$.

$$ \begin{aligned} \frac{y^2-4y-5}{y^2+5y+4} & =\frac{ \left( y-5 \right) \cdot \color{blue}{ \left( y+1 \right) }}{ \left( y+4 \right) \cdot \color{blue}{ \left( y+1 \right) }} = \\[1ex] &= \frac{y-5}{y+4} \end{aligned} $$

Multiply $ \dfrac{y-5}{y+4} $ by $ \dfrac{y^2+4y}{y^2-25} $ to get $ \dfrac{ y }{ y+5 } $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{y-5}{y+4} \cdot \frac{y^2+4y}{y^2-25} & \xlongequal{\text{Step 1}} \frac{ 1 \cdot \color{blue}{ \left( y-5 \right) } }{ 1 \cdot \color{red}{ \left( y+4 \right) } } \cdot \frac{ y \cdot \color{red}{ \left( y+4 \right) } }{ \left( y+5 \right) \cdot \color{blue}{ \left( y-5 \right) } } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 1 }{ 1 } \cdot \frac{ y }{ y+5 } \xlongequal{\text{Step 3}} \frac{ 1 \cdot y }{ 1 \cdot \left( y+5 \right) } \xlongequal{\text{Step 4}} \frac{ y }{ y+5 } \end{aligned} $$