The straightforward approach to finding P(-2) is to replace all the x's with -2's and then simplify. But this process can be tedious: raising -2 to various powers and then do all that adding and /or subtracting.
Synthetic division is a quick, fairly simple way to divide a polynomial by something of the form (x-a). And the Remainder Theorem tells us that for any polynomial, P(x), P(a) will be the remainder of . These facts combine to explain why we can use synthetic division to find the value of a polynomial. It is often much easier this way to find P(a) than the straightforward approach described above.
Probably the easiest way to get this wrong is to fail to notice that there is no term. When we set up the synthetic division we must notice this and know to use a 0 for its coefficient:
-2 | 1 9 0 -9 2
=== -2 -14 28 -38
=====================
1 7 -14 19 -36
The remainder of this division is always in the lower right corner. So your remainder, and therefore P(-2), is -36.