Subtract $ \dfrac{2}{t-1} $ from $ \dfrac{t}{t+1} $ to get $ \dfrac{ \color{purple}{ t^2-3t-2 } }{ t^2-1 }$.

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-1 }$ and the second by $\color{blue}{ t+1 }$.

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