Help with a cubic function problem?

We aren’t supposed to tell you the answer since this appears to be homework, but I will give you advice.

Whenever you have a question involving translations, what you should do is to graph the starting equation (in this case x^3), and then perform all of the translation on the graph, making a new sketch after each transformation. Then, what you do is to take the new graph and then write the formula for it, giving you your answer.

Regarding your answer, it does not seem to be correct as is.

If you replace the -3 with the +3, it will be correct. The rule for vertical movement is the opposite of vertical movement. To move horizontally by 2, you correctly subtracted 2. To move 3 vertically you have to add 3.

You seem to have gotten most of it, so I’ll confirm:
Reflection by x-axis: take the negative of y (done)
Vertical stretch by 2: double the value of y (done)
Translate 2 to right: subtract 2 from x (done)
Translate 3 up: add 3 to y (unsure of correct answer)

Fellow jellies will swarm me with tentacles because we never give homework help at this q&a site though imho at least one worked mathematical example, to get the OP on track, is in the spirit of hint, not answer & I like answering math questions by giving hints if I can.

@kittykat219 You had 3–½ out of 4 parts already done correctly & demonstrated basic concepts, so you’re there! Realize that the graph of y = f(x) will translate upward by c units by adding a positive quantity c (such as c=3) to y=f(x) to make y=f(x)+c. Similarly the graph of y=f(x-c) translates c units to the right. You can figure out y=cf(x) and y = f(cx) on your own.

Translations (shifts) & dilatations (scaling factors) are part of so-called affine geometry, which applies a set of linear transformations to geometric objects – a routine part of (for instance) image processing.

I should point out that the distinction between how to treat vertical and horizontal movement disappears if the 3 is combined with the y:
(y-3) = -2(x-3)^3, which then becomes y = -2(x-3)^3 + 3
Thinking of it in this way should make it easier to remember.