Synthetic division is a shortcut for dividing polynomials of the form x-a among others. It looks like long division by a, except there are no carries and we add instead of subtract.
This one tells us
[tex]\dfrac{2x^3 - 6x^2 + 3x - 30}{x - 4} = 2x^2 + 2x + 11 + \dfrac{14}{x-4}[/tex]
By the polynomial remainder theorem, that remainder of 14 is f(4). Let's check it
[tex]2(4)^3 - 6(4)^2 + 3(4) - 30 = 14[/tex]
Wow, math works.
For part 2, it's a bit of a trick question. When f(x) is divided by x-3 the remainder will be f(3) not f(-3).
That doesn't really tell us anything about f(-3); we'd have to evaluate it or divide by x+3 to make sure it isn't 15, but it very likely isn't.
I'll assume by separate problem they mean a different, unspecified f.
