Multiply each term of $ \left( \color{blue}{t-2}\right) $ by each term in $ \left( t-1\right) $.

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

Combine like terms:

$$ t^2 \color{blue}{-t} \color{blue}{-2t} +2 = t^2 \color{blue}{-3t} +2 $$

Multiply $ \color{blue}{6} $ by $ \left( 2t+1\right) $ $$ \color{blue}{6} \cdot \left( 2t+1\right) = 12t+6 $$

Find $ \left(t-1\right)^2 $ using formula.

$$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$

where $ A = \color{blue}{ t } $ and $ B = \color{red}{ 1 }$.

$$ \begin{aligned}\left(t-1\right)^2 = \color{blue}{t^2} -2 \cdot t \cdot 1 + \color{red}{1^2} = t^2-2t+1\end{aligned} $$

Multiply $ \color{blue}{2} $ by $ \left( 2t+1\right) $ $$ \color{blue}{2} \cdot \left( 2t+1\right) = 4t+2 $$

Multiply $t^2-3t+2$ by $ \dfrac{2t-3}{12t+6} $ to get $ \dfrac{2t^3-9t^2+13t-6}{12t+6} $.

Step 1: Write $ t^2-3t+2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} t^2-3t+2 \cdot \frac{2t-3}{12t+6} & \xlongequal{\text{Step 1}} \frac{t^2-3t+2}{\color{red}{1}} \cdot \frac{2t-3}{12t+6} \xlongequal{\text{Step 2}} \frac{ \left( t^2-3t+2 \right) \cdot \left( 2t-3 \right) }{ 1 \cdot \left( 12t+6 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2t^3-3t^2-6t^2+9t+4t-6 }{ 12t+6 } = \frac{2t^3-9t^2+13t-6}{12t+6} \end{aligned} $$

Multiply $3$ by $ \dfrac{t^2-2t+1}{2t+1} $ to get $ \dfrac{ 3t^2-6t+3 }{ 2t+1 } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $t$ by $ \dfrac{2t^2+2t+1}{4t+2} $ to get $ \dfrac{ 2t^3+2t^2+t }{ 4t+2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2t^3-9t^2+13t-6}{12t+6} $ and $ \dfrac{3t^2-6t+3}{2t+1} $ to get $ \dfrac{ \color{purple}{ 2t^3+9t^2-23t+12 } }{ 12t+6 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{6}$.

$$ \begin{aligned} \frac{2t^3-9t^2+13t-6}{12t+6} + \frac{3t^2-6t+3}{2t+1} & = \frac{ 2t^3-9t^2+13t-6 }{ 12t+6 } + \frac{ \left( 3t^2-6t+3 \right) \cdot \color{blue}{ 6 }}{ \left( 2t+1 \right) \cdot \color{blue}{ 6 }} = \\[1ex] &=\frac{ \color{purple}{ 2t^3-9t^2+13t-6 } }{ 12t+6 } + \frac{ \color{purple}{ 18t^2-36t+18 } }{ 12t+6 } = \\[1ex] &=\frac{ \color{purple}{ 2t^3+9t^2-23t+12 } }{ 12t+6 } \end{aligned} $$

Multiply $t$ by $ \dfrac{2t^2+2t+1}{4t+2} $ to get $ \dfrac{ 2t^3+2t^2+t }{ 4t+2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2t^3+9t^2-23t+12}{12t+6} $ and $ \dfrac{2t^3+2t^2+t}{4t+2} $ to get $ \dfrac{ \color{purple}{ 8t^3+15t^2-20t+12 } }{ 12t+6 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{2t^3+9t^2-23t+12}{12t+6} + \frac{2t^3+2t^2+t}{4t+2} & = \frac{ 2t^3+9t^2-23t+12 }{ 12t+6 } + \frac{ \left( 2t^3+2t^2+t \right) \cdot \color{blue}{ 3 }}{ \left( 4t+2 \right) \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 2t^3+9t^2-23t+12 } }{ 12t+6 } + \frac{ \color{purple}{ 6t^3+6t^2+3t } }{ 12t+6 } = \\[1ex] &=\frac{ \color{purple}{ 8t^3+15t^2-20t+12 } }{ 12t+6 } \end{aligned} $$