You need more practice with matrix multiplication! If you multiply that matrix with the vector (x,y,1), you get (2x,3y,1) -- which is obviously not what you want. You want the result to be (2x+2, 2y+3, 1).
The articles I linked to say that scaling by a factor of k looks like this:
Translation by (p,q) looks like this:
Multiply them both out to confirm that this works.
Now, to combine them you can multiply the matrices. If you do A first, then B, then you need it to look like this: B(Av) because you are applying A first, to get Av, then applying B to get B(Av). So the matrix which does both in one go is BA. (If you did B first, then A, it would be AB.)
You will find that the resulting matrix BA (when A is the scaling, and B is the translation) is pretty simple.
Multiply out the right-hand side to check that it works.
Also, check if you are supposed to use homogeneous coordinates in the exam. Maybe they have something different in mind.