Simplify $ \dfrac{6p^2-13p+5}{2p^2+17p-9} $ to $ \dfrac{3p-5}{p+9} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{2p-1}$.

$$ \begin{aligned} \frac{6p^2-13p+5}{2p^2+17p-9} & =\frac{ \left( 3p-5 \right) \cdot \color{blue}{ \left( 2p-1 \right) }}{ \left( p+9 \right) \cdot \color{blue}{ \left( 2p-1 \right) }} = \\[1ex] &= \frac{3p-5}{p+9} \end{aligned} $$

Multiply $ \dfrac{3p-5}{p+9} $ by $ \dfrac{p^2+16p+63}{4} $ to get $ \dfrac{3p^2+16p-35}{4} $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3p-5}{p+9} \cdot \frac{p^2+16p+63}{4} & \xlongequal{\text{Step 1}} \frac{ 3p-5 }{ 1 \cdot \color{red}{ \left( p+9 \right) } } \cdot \frac{ \left( p+7 \right) \cdot \color{red}{ \left( p+9 \right) } }{ 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 3p-5 }{ 1 } \cdot \frac{ p+7 }{ 4 } \xlongequal{\text{Step 3}} \frac{ \left( 3p-5 \right) \cdot \left( p+7 \right) }{ 1 \cdot 4 } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 3p^2+21p-5p-35 }{ 4 } = \frac{3p^2+16p-35}{4} \end{aligned} $$

Multiply $ \dfrac{3p^2+16p-35}{4} $ by $ p $ to get $ \dfrac{ 3p^3+16p^2-35p }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3p^2+16p-35}{4} \cdot p & \xlongequal{\text{Step 1}} \frac{3p^2+16p-35}{4} \cdot \frac{p}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 3p^2+16p-35 \right) \cdot p }{ 4 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3p^3+16p^2-35p }{ 4 } \end{aligned} $$

Add $ \dfrac{3p^3+16p^2-35p}{4} $ and $ 28 $ to get $ \dfrac{ \color{purple}{ 3p^3+16p^2-35p+112 } }{ 4 }$.

Step 1: Write $ 28 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{4}$.

$$ \begin{aligned} \frac{3p^3+16p^2-35p}{4} +28 & \xlongequal{\text{Step 1}} \frac{3p^3+16p^2-35p}{4} + \frac{28}{\color{red}{1}} = \frac{ 3p^3+16p^2-35p }{ 4 } + \frac{ 28 \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3p^3+16p^2-35p } }{ 4 } + \frac{ \color{purple}{ 112 } }{ 4 }=\frac{ \color{purple}{ 3p^3+16p^2-35p+112 } }{ 4 } \end{aligned} $$