Divide $ \dfrac{2r^2+11r-6}{r^2} $ by $ 4r+5 $ to get $ \dfrac{ 2r^2+11r-6 }{ 4r^3+5r^2 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{2r^2+11r-6}{r^2} }{4r+5} & \xlongequal{\text{Step 1}} \frac{2r^2+11r-6}{r^2} \cdot \frac{\color{blue}{1}}{\color{blue}{4r+5}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 2r^2+11r-6 \right) \cdot 1 }{ r^2 \cdot \left( 4r+5 \right) } \xlongequal{\text{Step 3}} \frac{ 2r^2+11r-6 }{ 4r^3+5r^2 } \end{aligned} $$

Divide $ \dfrac{2r^2+11r-6}{4r^3+5r^2} $ by $ r $ to get $ \dfrac{ 2r^2+11r-6 }{ 4r^4+5r^3 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{2r^2+11r-6}{4r^3+5r^2} }{r} & \xlongequal{\text{Step 1}} \frac{2r^2+11r-6}{4r^3+5r^2} \cdot \frac{\color{blue}{1}}{\color{blue}{r}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 2r^2+11r-6 \right) \cdot 1 }{ \left( 4r^3+5r^2 \right) \cdot r } \xlongequal{\text{Step 3}} \frac{ 2r^2+11r-6 }{ 4r^4+5r^3 } \end{aligned} $$

Add $ \dfrac{2r^2+11r-6}{4r^4+5r^3} $ and $ 2 $ to get $ \dfrac{ \color{purple}{ 8r^4+10r^3+2r^2+11r-6 } }{ 4r^4+5r^3 }$.

Step 1: Write $ 2 $ 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 $\color{blue}{4r^4+5r^3}$.

$$ \begin{aligned} \frac{2r^2+11r-6}{4r^4+5r^3} +2 & \xlongequal{\text{Step 1}} \frac{2r^2+11r-6}{4r^4+5r^3} + \frac{2}{\color{red}{1}} = \frac{ 2r^2+11r-6 }{ 4r^4+5r^3 } + \frac{ 2 \cdot \color{blue}{ \left( 4r^4+5r^3 \right) }}{ 1 \cdot \color{blue}{ \left( 4r^4+5r^3 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2r^2+11r-6 } }{ 4r^4+5r^3 } + \frac{ \color{purple}{ 8r^4+10r^3 } }{ 4r^4+5r^3 } = \\[1ex] &=\frac{ \color{purple}{ 8r^4+10r^3+2r^2+11r-6 } }{ 4r^4+5r^3 } \end{aligned} $$