Let's perform the synthetic division

#color(white)(aaaa)##3##|##color(white)(aaaa)##1##color(white)(aaaa)##-7##color(white)(aaaaaa)##5##color(white)(aaaaaa)##-6#

#color(white)(aaaaa)####|##color(white)(aaaa)####color(white)(aaaaaa)##3##color(white)(aaaaa)##-12##color(white)(aaaa)##-21#

#color(white)(aaaaaaaaa)###_________________________________________________________##

#color(white)(aaaaa)####|##color(white)(aaaa)##1##color(white)(aaaa)##-4##color(white)(aaaaaa)##-7##color(white)(aaaaaa)##color(red)(-27)#

The remainder is #-27# and the quotient is #=x^2-4x-7#

Therefore,

#(x^3-7x^2+5x-6)/(x-3)=(x^2-4x-7)-27/(x-3)#