Multiply $ \dfrac{15}{4} $ by $ \dfrac{t^2}{t} $ to get $ \dfrac{ 15t^2 }{ 4t } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{15}{4} \cdot \frac{t^2}{t} \xlongequal{\text{Step 1}} \frac{ 15 \cdot t^2 }{ 4 \cdot t } \xlongequal{\text{Step 2}} \frac{ 15t^2 }{ 4t } \end{aligned} $$

Subtract $ \dfrac{15t^2}{4t} $ from $ t^2+2t $ to get $ \dfrac{ \color{purple}{ 4t^3-7t^2 } }{ 4t }$.

Step 1: Write $ t^2+2t $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} t^2+2t- \frac{15t^2}{4t} & \xlongequal{\text{Step 1}} \frac{t^2+2t}{\color{red}{1}} - \frac{15t^2}{4t} = \frac{ \left( t^2+2t \right) \cdot \color{blue}{ 4t }}{ 1 \cdot \color{blue}{ 4t }} - \frac{ 15t^2 }{ 4t } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4t^3+8t^2 } }{ 4t } - \frac{ \color{purple}{ 15t^2 } }{ 4t }=\frac{ \color{purple}{ 4t^3-7t^2 } }{ 4t } \end{aligned} $$

Subtract $ \dfrac{3}{t} $ from $ \dfrac{4t^3-7t^2}{4t} $ to get $ \dfrac{ \color{purple}{ 4t^4-7t^3-12t } }{ 4t^2 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ t }$ and the second by $\color{blue}{ 4t }$.

$$ \begin{aligned} \frac{4t^3-7t^2}{4t} - \frac{3}{t} & = \frac{ \left( 4t^3-7t^2 \right) \cdot \color{blue}{ t }}{ 4t \cdot \color{blue}{ t }} - \frac{ 3 \cdot \color{blue}{ 4t }}{ t \cdot \color{blue}{ 4t }} = \\[1ex] &=\frac{ \color{purple}{ 4t^4-7t^3 } }{ 4t^2 } - \frac{ \color{purple}{ 12t } }{ 4t^2 }=\frac{ \color{purple}{ 4t^4-7t^3-12t } }{ 4t^2 } \end{aligned} $$