#(8x^3 + 12x^2 - 6x + 8)/(2x+6)#
dividing numerator and denominator by 2 (purely for simplification purpose)
= #(4x^3 + 6x^2 - 3x + 4)/(x+3)#
try to break the polynomial in the numerator such that we can factor out the denominator from successive terms
= #(4x^3 + 12x^2 - 6x^2 - 18x + 15x + 45 - 41)/(x+3)#2
=#(4x^2(x+3) - 6x(x+3) +15(x+3) - 41)/(x+3)#
=#(4x^2(x+3) - 6x(x+3) +15(x+3))/(x+3) - 41/(x+3)#
= #((x+3)(4x^2 - 6x + 15))/(x+3) - 41/(x+3) #
=#(4x^2-6x+15) - 41/(x+3) #
the second term in the expression #i.e. - 41/(x+3)# cannot be simplified further as degree of numerator is less than that of denominator.
[Degree is the highest power of variable in a polynomial
(#therefore# degree of #-41# is #0# and that of #x+3# is #1# ) ]
#therefore# #(8x^3 + 12x^2 - 6x + 8)/(2x+6) = (4x^2-6x+15) - 41/(x+3)#
or
quotient #= (4x^2-6x+15) # & remainder #= 2*(-41) = -82# since we had initially divided both numerator and denominator by 2, we have to multiply -41 by 2 to get the correct remainder.