How do you find the quotient and remainder of p of x equals x to the fourth plus 2x cubed plus x squared plus 8x minus 12 divided by x plus 2?

P(x) is a polynomial of order 4 and you are dividing by a polynomial of order 1 so the quotient will be of order 4 - 1 = 3

So suppose the quotient is Ax3 + Bx2 + Cx + D

Then p(x)/(x + 2) = Ax3 + Bx2 + Cx + D with remainder R.

To find R, simply evaluate p(x) at x = -2.

p(2) = -24

Cross-multiply:

p(x) = (x + 2)*(Ax3 + Bx2 + Cx + D) - 24

= Ax4 + 2Ax3 + Bx3 + 2Bx2 + Cx2 + 2Cx + Dx + 2D - 24

Comparing coefficients of:

x4: 1 = A

x3: 2 = 2A +B = 2 + B => B = 0

x2: 1 = 2B + C = 0 + C => C = 1

x : 8 = 2C + D = 2 + D => D = 6

and, as a check,

x0 : -12 = 2D + R = 12 + R => R = -24