After you clear the fraction, you will have an equation where $n$ appears on both sides. Move all terms involving $n$ to the same side of the equation.

Once all $n$ terms are on one side, factor out $n$ and see what is multiplying it. Then divide both sides by that factor to isolate $n$.

Be careful with negative signs when you rearrange terms like $1 - k^2$ and $k^2 - 1$. They are not the same; one is the negative of the other.

Choose a simple value for $k$ that does not cause division by zero in the answer choices (avoid $k = \pm 1$). For example, type k = 2 in Desmos to fix a specific value of $k$ and note that $k^2$ is then $4$.

For your chosen $k$, enter each option’s expression in Desmos as n_A = 4*k^2/(1 - k^2), n_B = 4*k^2/(k^2 - 1), n_C = 4*k^2*(k^2 - 1), and n_D = (k^2 - 1)/(4*k^2). Desmos will give you a numeric value for each $n_\text{choice}$.

For each computed value, enter n_A/(n_A - 4), n_B/(n_B - 4), etc., and compare each result to k^2. The choice whose $n_\text{choice}$ value makes n/(n - 4) equal to k^2 for your test $k$ is the correct formula for $n$ in terms of $k$.

Start with the equation

Multiply both sides by $n - 4$ to get rid of the denominator:

Distribute $k^2$ on the right-hand side:

Now get all the $n$ terms on one side by subtracting $k^2 n$ from both sides:

Factor $n$ out of the left-hand side:

It is often easier to work without a leading negative sign. Multiply both sides of

by $-1$:

Now $n$ is multiplied by $(k^2 - 1)$, so the last step is to divide by this factor.

Divide both sides of

by $k^2 - 1$ (noting $k^2 \neq 1$ so we are not dividing by zero):

This matches answer choice B.