Expand (x+1)(4x+12)(x+1)(4x+12) into 4x^2+16x+124x2+16x+12
Now you can long divide (-3x^3+12x^2-7x-6)/(4x^2+16x+12)−3x3+12x2−7x−64x2+16x+12.
First, divide the leading coefficients, that is -3x^3−3x3 and 4x^24x2 into -3x/4−3x4. Now multiply (4x^2+16x+12)(4x2+16x+12) by -3x/4−3x4 to get -3x^3-12x^2-9x−3x3−12x2−9x. Subtract -3x^3-12x^2-9x−3x3−12x2−9x from -3x^3+12x^2-7x-6−3x3+12x2−7x−6 to get the remainder, that is 24x^2+2x-624x2+2x−6.
Therefore,
(-3x^3+12x^2-7x-6)/(4x^2+16x+12)=-3x/4+(24x^2+2x-6)/(4x^2+16x+12)−3x3+12x2−7x−64x2+16x+12=−3x4+24x2+2x−64x2+16x+12.
Now, divide (24x^2+2x-6)/(4x^2+16x+12)24x2+2x−64x2+16x+12 with the same steps above. First, divide the leading coefficients to get 66. Multiply 4x^2+16x+124x2+16x+12 by 66 to get 24x^2+96x+7224x2+96x+72. Subtract 24x^2+96x+7224x2+96x+72 from 24x^2+2x-624x2+2x−6 to get the second remainder, -94x-78−94x−78.
Therefore,
(24x^2+2x-6)/(4x^2+16x+12)=6+(-94x-78)/(4x^2+16x+12)24x2+2x−64x2+16x+12=6+−94x−784x2+16x+12.
Adding together gives us
-3x/4+6+(-94x-78)/(4x^2+16x+12)−3x4+6+−94x−784x2+16x+12
which simplifies into
-3x/4+6-(47x+39)/(2(x^2+4x+3))−3x4+6−47x+392(x2+4x+3)
