We know that,
Dividend-:÷ Divisor = quotient and remainder.In short,
Dividend = Divisor xx× quotient + remainder.
Now,divisor is (x+2),and
dividend=2x^3+3x^2-4x-2=2x3+3x2−4x−2=2x^3+4x^2-x^2-2x-2x-4+2=2x3+4x2−x2−2x−2x−4+2
=2x^2(x+2)-x(x+2)-2(x+2)+color(red)(2)=2x2(x+2)−x(x+2)−2(x+2)+2
=(x+2)(2x^2-x-2)+2=(x+2)(2x2−x−2)+2
i.e.quotient=(2x^2-x-2)(2x2−x−2),and remainder=2.
So,
2x^3+3x^2-4x-2=(x+2)(2x^2-x-2)+22x3+3x2−4x−2=(x+2)(2x2−x−2)+2
Hence,
(2x^3+3x^2-4x-2)/(x+2)=[(x+2)(2x^2-x-2)+2]/(x+2)2x3+3x2−4x−2x+2=(x+2)(2x2−x−2)+2x+2
=((x+2)(2x^2-x-2))/(x+2)+2/(x+2)=(x+2)(2x2−x−2)x+2+2x+2
=(2x^2-x-2)+color(red)(2/(x+2)=(2x2−x−2)+2x+2
