Multiply $ \color{blue}{5} $ by $ \left( p+3\right) $ $$ \color{blue}{5} \cdot \left( p+3\right) = 5p+15 $$

Multiply $ \color{blue}{25} $ by $ \left( p+4\right) $ $$ \color{blue}{25} \cdot \left( p+4\right) = 25p+100 $$

Multiply each term of $ \left( \color{blue}{25p+100}\right) $ by each term in $ \left( p-1\right) $.

$$ \left( \color{blue}{25p+100}\right) \cdot \left( p-1\right) = 25p^2-25p+100p-100 $$

Combine like terms:

$$ 25p^2 \color{blue}{-25p} + \color{blue}{100p} -100 = 25p^2+ \color{blue}{75p} -100 $$

Multiply $5p+15$ by $ \dfrac{p+1}{25p^2+75p-100} $ to get $ \dfrac{p^2+4p+3}{5p^2+15p-20} $.

Step 1: Write $ 5p+15 $ 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.

$$ \begin{aligned} 5p+15 \cdot \frac{p+1}{25p^2+75p-100} & \xlongequal{\text{Step 1}} \frac{5p+15}{\color{red}{1}} \cdot \frac{p+1}{25p^2+75p-100} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( p+3 \right) \cdot \color{blue}{5} }{ 1 } \cdot \frac{ p+1 }{ \left( 5p^2+15p-20 \right) \cdot \color{blue}{5} } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ p+3 }{ 1 } \cdot \frac{ p+1 }{ 5p^2+15p-20 } \xlongequal{\text{Step 4}} \frac{ \left( p+3 \right) \cdot \left( p+1 \right) }{ 1 \cdot \left( 5p^2+15p-20 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ p^2+p+3p+3 }{ 5p^2+15p-20 } = \frac{p^2+4p+3}{5p^2+15p-20} \end{aligned} $$