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

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

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Combine like terms:

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{t^2-3t+2}{4t+2} $ and $ \dfrac{2t-2}{2t+1} $ to get $ \dfrac{ \color{purple}{ t^2+t-2 } }{ 4t+2 }$.

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

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

Add $ \dfrac{t^2+t-2}{4t+2} $ and $ \dfrac{t+1}{2} $ to get $ \dfrac{ \color{purple}{ 6t^2+8t-2 } }{ 8t+4 }$.

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

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