Express (1)/((3t+1)(t+1)) in partial fractions.

Step 1: Express $\frac{1}{(3t+1)(t+1)}$ in partial fractions.
We set up the partial fraction decomposition as:

$$\frac{1}{(3t+1)(t+1)} = \frac{A}{3t+1} + \frac{B}{t+1}$$

Multiply both sides by $(3t+1)(t+1)$ to clear the denominators:

$$1 = A(t+1) + B(3t+1)$$

To find $A$ and $B$, we can substitute values for $t$.
Let $t = -1$:

$$1 = A(-1+1) + B(3(-1)+1)$$ $$$$