Write a new function that represents the reflection of $f(x)$ across the $x$-axis. Start by writing $-f(x)$ and then substitute $f(x)=\sqrt{x+4}-3$.

Once you have the reflected function $h(x)$, a shift 2 units to the right means you should replace $x$ with what expression inside $h$?

After you substitute into $h(x)$ for the rightward shift, simplify the expression inside the square root and compare your final equation to the answer choices.

Type f(x)=sqrt(x+4)-3 to graph the given function $f(x)$.

On a new line, type r(x)=-f(x) to graph the reflection of $f(x)$ across the $x$-axis. Observe how the graph flips over the $x$-axis and note the new equation shown by Desmos for $r(x)$.

On another line, type g(x)=r(x-2) (or equivalently g(x)=-f(x-2)) to shift the reflected graph 2 units to the right. Look at the equation Desmos displays for g(x) and compare that equation to the answer choices to see which one matches.

The function is

You are told that $g(x)$ is obtained from the graph of $f(x)$ by:

1. Reflecting it across the $x$-axis.
2. Then translating (shifting) it 2 units to the right.

We will apply these transformations in this order to the equation of $f(x)$.

Reflecting a graph across the $x$-axis changes each $y$-value to its opposite, so algebraically you replace $f(x)$ with $-f(x)$.

So the reflected function is

Substitute $f(x)=\sqrt{x+4}-3$:

This is the equation after reflection but before the horizontal shift.

To translate a graph 2 units to the right, you replace $x$ with $x-2$ in the function.

We now apply this to the reflected function $h(x)=-\sqrt{x+4}+3$:

Now simplify the expression inside the square root:

so

This matches answer choice C.