We need to use long division to find our answer.
Step 1: #2x# goes into #x^4#, #.5x^3# times, so we need to multiply our divisor, #2x+1#, by #.5x^3#, and subtract that from the dividend, #x^4 + 4x^3 - 12x^2 - 10x + 2#.
#.5x^3 (2x+1) = x^4 + .5x^3#
#(x^4 + 4x^3 - 12x^2 - 10x + 2) - (x^4 + .5x^3) = 3.5x^3 - 12x^2 - 10x + 2#
Step 2: #2x# goes into #3.5x^3#, #1.75x^2# times. Repeat step 1 with these values.
#1.75x^2(2x+1) = 3.5x^3 + 1.75x^2#
#(3.5x^3 - 12x^2 - 10x + 2) - (3.5x^3 + 1.75x^2) = 13.75x^2 - 10x + 2#
Step 3: #2x# into #13.75x^2#, #6.875x# times. Repeat step 1.
#6.875x(2x+1) = 13.75x^2 + 6.875x#
#(13.75x^2 - 10x + 2) - (13.75x^2 + 6.875x) = -16.875x + 2#
Step 4: #2x# into #-16.875x#, #-8.4375# times. Repeat step 1.
#-8.4375(2x+1) = -16.875x - 8.4375#
#(-16.875x + 2) - (-16.875x - 8.4375) = 10.4375#
#2x# can't go into 10.4375, so it is a remainder. Taking our divisors and remainder, our answer is:
#.5x^3 + 1.75x^2 + 6.875x - 8.4375 + (10.4375)/(2x+1)#
