Write as a long division problem:
#color(white)("XXXX")underline(color(white)("XXXXXXXXXX")#
#7x+3 ) 4x^2-17x-11#
Divide the leading term of the divisor into the leading term of the dividend to get the first term of the quotient.
#color(white)("XXXXX")underline(color(red)(4/7x)color(white)("XXXXXXX")#
#color(red)(7x)+3) 4x^2-17x-11#
Multiply the entire divisor by the first term of the quotient
and subtract this from the original dividend.
#color(white)("XXXX")underline(color(red)(4/7x)color(white)("XXXXXXX")#
#color(red)(7x+3) ) 4x^2-17x-11#
#color(white)("XXXX")underline(color(red)(4x^2-12/7xcolor(white)("XXX")))#
#color(white)("XXXXXXX")color(blue)(-107/7x-11)#
Divide the first term of the divisor into the first term of the new dividend/remainder to get the second term of the quotient.
#color(white)("XXXX")underline(4/7xcolor(red)(-107/49)color(white)("XXX")#
#color(red)(7x)+3 ) 4x^2-17x-11#
#color(white)("XXXX")underline(4x^2-12/7xcolor(white)("XXX"))#
#color(white)("XXXXXXX")color(red)(-107/7x)-11#
Multiply the entire divisor by this second term in the quotient and subtract from the dividend/remainder.
#color(white)("XXXX")underline(4/7xcolor(red)(-107/49)color(white)("XXX")#
#color(red)(7x+3) ) 4x^2-17x-11#
#color(white)("XXXX")underline(4x^2-12/7xcolor(white)("XXX"))#
#color(white)("XXXXXXX")-107/7x-11#
#color(white)("XXXXXXX")underline(color(red)(-107/7x-321/49))#
#color(white)("XXXXXXXXXXXX")color(blue)(-218/49)#
#color(white)("XXXX")underline(4/7x-107/49color(white)("XXX")#(Quotient)
#7x+3 ) 4x^2-17x-11#
#color(white)("XXXX")underline(4x^2-12/7xcolor(white)("XXX"))#
#color(white)("XXXXXXX")-107/7x-11#
#color(white)("XXXXXXX")underline(-107/7x-321/49)#
#color(white)("XXXXXXXXXXXX")-218/49#(Remainder)
