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(x+1)(x+10)=x2+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))x2+11x+10x2+11x+10)x3+4x2−7x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x3+4x2−7x−6
x^2x2 goes into x^3x3 xx 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))x2+11x+10x2+11x+10)x+4x2−7x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x3+4x2−7x−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))x2+11x+10x2+11x+10)x3+11x2+10x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x3−7x2−17x−6
x^2x2 goes into -7x^2−7x2 -7−7 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))x2+11x+10x2+11x+10)x−7x2−7x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x3+4x2−7x−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))x2+11x+10x2+11x+10)x3+11x2+10x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x3−7x2−17x−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))x2+11x+10x2+11x+10)x3−7x2−77x−70¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x3+0x2+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))x3+4x2−7x−6x2+11x+10=(x−7)+60x+64x2+11x+10=(x−7)+4(15x+16)(x+1)(x+10)
