If you let $t = \dfrac{y}{x-2}$ (or $t = \dfrac{x-2}{y}$), rewrite both terms in the equation using $t$ and then multiply to clear the denominator.

After substituting and clearing denominators, you should get a quadratic equation in $t$. Use the quadratic formula and then convert back from $t$ to $y$.

In Desmos, type the expression ((x-2)/y) + (y/(x-2)) and note that valid $(x,y)$ pairs must make this equal to 5. You will verify each answer choice by substitution rather than solving symbolically.

For each option, define functions of $x$ using the + sign first. For example, for choice A define yA(x) = (x-2)*(5+sqrt(21))/2; for choice B define yB(x) = (x-2)*(5+sqrt(17))/2; similarly define yC(x) and yD(x) from their formulas (using the + in place of ±).

For each candidate function, define a new function that plugs it into the original expression, e.g., fA(x) = (x-2)/yA(x) + yA(x)/(x-2). Then create a table of values for $x$ (such as $x=0,1,3,4$, avoiding $x=2$) and look at $fA(x)$, $fB(x)$, etc. The correct answer choice will be the one whose corresponding function consistently gives $5$ for all tested $x$-values.

Start with

where $x\ne 2$ and $y\ne 0$.

Let

Then

so the equation becomes

Multiply both sides by $t$ (allowed because $t\ne 0$):

which rearranges to the quadratic

Use the quadratic formula on

where $a=1$, $b=-5$, and $c=1$.

The quadratic formula gives

Substitute $a$, $b$, and $c$:

So

From

solve for $y$ by multiplying both sides by $(x-2)$:

This matches choice A) $y = \dfrac{(x-2)(5\pm\sqrt{21})}{2}$.