To simplify the problem, divide dividend and divisor by their common factor 3:
(4x3+x2-36)/(x-2) can be solved by synthetic division:
2 | 4 1 0 -36
4 8 18 | 36
4 9 18 | 0 = 4x2+9x+18.
Using algebraic division:
4x2+9x+18
x-2 ) 4x3 +x2 -36
4x3-8x2
9x2
9x2-18x
18x-36
18x-36
0