#color(white)((000)/color(black)(x^2+4)(000)/color(black)(")"bar(-3x^4-9x^2+2x-3)))#
First, remember to fill in any powers we don't have in the divisor and dividend with a #0x# to that power
#color(white)((000)/color(black)(x^2+0x+4)(000)/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
Now divide the first term in the dividend with the first term in the divisor
(#-3x^4-:x^2)#
[Notice how the result from the division is written above its corresponding power #(x^2)#]
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(000)color(black)(-3x^2)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
Now multiply our divisor #(x^2+0x+4)# with the result of our division #(-3x^2)# and write it like so...
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(000)color(black)(-3x^2)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000-3x^4-0x^3-12x^2#
Remember just like regular long division, we subtract the result of our multiplication so the signs change
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(000)color(black)(-3x^2)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000+3x^4+0x^3+12x^2#
Now we get...
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(000)color(black)(-3x^2)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000+3x^4+0x^3+12x^2#
#color(white)00000000000bar(color(white)(0000)0x^4+0x^3+3x^2)#
Bring down our remaining terms
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(000)color(black)(-3x^2)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000+3x^4+0x^3+12x^2#
#color(white)00000000000bar(color(white)(000000000000000)3x^2)+2x-3#
Now divide the first term from the subtraction with the first term from the divisor, kind of like from regular long division #(3x^2-:x^2)#
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(0000000)color(black)(-3x^2+3)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000+3x^4+0x^3+12x^2#
#color(white)00000000000bar(color(white)(000000000000000)3x^2)+2x-3#
and again, multiply the divisor #(x^2+0x+4)# with the result from our division #(3)#
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(0000000)color(black)(-3x^2+3)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000+3x^4+0x^3+12x^2#
#color(white)00000000000bar(color(white)(000000000000000)3x^2)+2x-3#
#color(white)(00000000000000000000000000)3x^2+0x+12#
[Remember we are subtracting so don't forget to change the signs!]
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(0000000)color(black)(-3x^2+3)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000+3x^4+0x^3+12x^2#
#color(white)00000000000bar(color(white)(000000000000000)3x^2)+2x-3#
#color(white)(000000000000000000000000)-3x^2-0x-12#
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(0000000)color(black)(-3x^2+3)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000+3x^4+0x^3+12x^2#
#color(white)00000000000bar(color(white)(000000000000000)3x^2)+2x-3#
#color(white)(000000000000000000000000)-3x^2-0x-12#
#color(white)(000000000000000000000000)bar(color(white)(000)0x^2+2x-15)#
#color(white)((000)/color(black)(x^2+0x+4)[color(white)(0000000)color(black)(-3x^2+3)]/color(black)(")"bar(-3x^4+0x^3-9x^2+2x-3)))#
#color(white)000000000000+3x^4+0x^3+12x^2#
#color(white)00000000000bar(color(white)(000000000000000)3x^2)+2x-3#
#color(white)(000000000000000000000000)-3x^2-0x-12#
#color(white)(000000000000000000000000)bar(color(white)(000000000)2x-15)#
Now, we are left with #2x-15#. Before you jump in and try to divide again, notice how we have gotten to the point where the first term in our divisor #(x^2)# is larger than the first term from our subtraction #(2x)#.
This means that #-3x^2+3# is our quotient and #2x-15# is our remainder.
When dividing polynomials, our answer is defined as the quotient #color(red)[(-3x^2+3)]#. Plus the remainder #color(green)[(2x-15)]# divided by the divisor #color(blue)[(x^2+4)]#
So...
#color(red)[-3x^2+3]+color(green)[2x-15]/color(blue)[x^2+4]#
