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