Simplify $ \dfrac{x^2-1}{x+1} $ to $ x-1$.

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

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

Multiply $x-1$ by $ \dfrac{x+5}{5x-5} $ to get $ \dfrac{ x+5 }{ 5 } $.

Step 1: Write $ x-1 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

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