Given: 6x³-9x²+5x+2 / 2x+3
Let f(x)=6x³-9x²+5x+2 ··· Ex.1
Plug x=-3/2 into Ex.1 We have: f(-3/2)=-46, so Ex.1 is indivisible by 2x+3, and the remainder is -46 if Ex.1 is devided by 2x+3.
Factor Ex.1 by 2x+3. The solution below is basically the same method as that used in long division.
6x³-9x²+5x+2=3x²(2x+3)-18x²+5x+2=3x²(2x+3)-9x(2x+3)+32x+2
=3x²(2x+3)-9x(2x+3)+16(2x+3)-46=(2x+3)(3x²-9x+16)-46
Thus, 6x³-9x²+5x+2 / 2x+3={(2x+3)(3x²-9x+16)-46} / 2x+3=(3x²-9x+16) R-46
The answer: the quotient is 3x²-9x+16, and the remainder is -46.