There are no factors of the numerator that can help us, and so we're left to do this via long division. Let's first expand the denominator:
#(x+1)(x+10)=x^2+11x+10#
And now for the long division:
#color(white)((x^2+11x+10)/color(black)(x^2+11x+10")")(x^3+4x^2-7x-6)/color(black)bar(x^3+4x^2-7x-6))#
#x^2# goes into #x^3# #x# times:
#color(white)((x^2+11x+10)/color(black)(x^2+11x+10")")(color(black)(x+)4x^2-7x-6)/color(black)bar(x^3+4x^2-7x-6))#
#color(white)((x^2+11x+10)/(x^2+11x+10")")(color(black)(x^3+11x^2+10xcolor(white)(-6)))/color(black)bar(0x^3-7x^2-17x-6))#
#x^2# goes into #-7x^2# #-7# times:
#color(white)((x^2+11x+10)/color(black)(x^2+11x+10")")(color(black)(x-7)x^2-7x-6)/color(black)bar(x^3+4x^2-7x-6))#
#color(white)((x^2+11x+10)/(x^2+11x+10")")(color(black)(x^3+11x^2+10xcolor(white)(-6)))/color(black)bar(color(white)(0x^3)-7x^2-17x-6))#
#color(white)((x^2+11x+10)/(x^2+11x+10")")(color(black)(color(white)(x^3)-7x^2-77x-70))/color(black)bar(color(white)(0x^3)+0x^2+60x+64))#
This gives us:
#(x^3+4x^2-7x-6)/(x^2+11x+10)=(x-7)+(60x+64)/(x^2+11x+10)=(x-7)+(4(15x+16))/((x+1)(x+10))#
