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]
