Once you know $m$, plug $x=5$ and $r(5)=2$ into $r(x)=\dfrac{mx+n}{x-3}$ to solve for $n$.

After you have $r(x)$, replace $x$ with $x-2$ to get $r(x-2)$, then add 1. Be sure to simplify step by step before comparing with the answer choices.

Write the "+1" in $s(x)=r(x-2)+1$ with the same denominator as $r(x-2)$ so you can combine them into a single rational expression like the answer choices.

In Desmos, enter $r(x) = (4x - 16)/(x - 3)$, using $m=4$ from the asymptote and $n=-16$ from the point $(5,2)$. Check that the graph approaches $y=4$ for large $|x|$ and that the point $(5,2)$ lies on the graph.

In a new line, define $s(x) = r(x - 2) + 1$. Observe that this graph is the graph of $r(x)$ shifted 2 units to the right and 1 unit up.

On separate lines, type each of the four answer choices exactly as shown in the problem, naming them for convenience (for example, $A(x)$, $B(x)$, etc.). Then compare their graphs to $s(x)$ and see which one coincides with $s(x)$ for all $x$ in the domain. The matching one corresponds to the correct equation for $s(x)$.

For a rational function of the form

both numerator and denominator are linear (degree 1). The horizontal asymptote is the ratio of the leading coefficients, so it is $y=\frac{m}{1}=m$.

We are told the horizontal asymptote is $y=4$, so

Now we know

The graph passes through $(5,2)$, so $r(5)=2$.

Multiply both sides by 2:

Substitute $m=4$ and $n=-16$ back into $r(x)$:

To get $r(x-2)$, replace $x$ with $x-2$ in the formula for $r$:

Simplify numerator and denominator:

So

Then

Write $1$ with denominator $x-5$ so you can combine the terms:

so

Therefore, the equation that could define $s$ is $s(x)=\dfrac{5x-29}{x-5}$ (choice C).