Once you have an equation involving $\frac{\nu}{s}$ and $\frac{\nu}{t}$, what can you multiply the whole equation by so that $s$ and $t$ no longer appear in denominators?

After clearing denominators, you will get an equation where $s$ appears in more than one term. Put all the terms with $s$ on one side, factor out $s$, and then divide to solve for $s$.

In Desmos, pick easy positive numbers for $t$ and $\nu$ that keep denominators nonzero, such as typing t = 3 and v = 2 (using v to stand for $\nu$).

For each option, define the corresponding expression for $s$ in Desmos, for example sA = (expression from choice A), sB = (expression from choice B), and so on, using $t$ and v in place of $\nu$.

For each defined $s$ (like sA, sB, etc.), create a line such as diffA = 2/(1/sA + 1/t) - v. The expression whose diff value is exactly 0 (or extremely close to 0) for your chosen $t$ and v is the one that correctly represents $s$ in terms of $t$ and $\nu$.

Begin with

Multiply both sides by the denominator $\frac{1}{s} + \frac{1}{t}$ to get rid of the fraction in the denominator:

From

distribute $\nu$:

Now multiply the entire equation by $st$ (the common denominator of $s$ and $t$) to clear the remaining fractions:

We now have

Move the $\nu s$ term to the right side so that all $s$ terms are together:

Factor $s$ out of the terms on the right:

From

divide both sides by $2t - \nu$ (which is allowed as long as $2t - \nu \neq 0$):

So $s$ expressed in terms of $t$ and $\nu$ is $s = \dfrac{\nu t}{2t - \nu}$, which corresponds to choice D.