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)=x2+11x+10
And now for the long division:
x2+11x+10x2+11x+10)x3+4x2−7x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x3+4x2−7x−6
x2 goes into x3 x times:
x2+11x+10x2+11x+10)x+4x2−7x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x3+4x2−7x−6
x2+11x+10x2+11x+10)x3+11x2+10x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x3−7x2−17x−6
x2 goes into −7x2 −7 times:
x2+11x+10x2+11x+10)x−7x2−7x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x3+4x2−7x−6
x2+11x+10x2+11x+10)x3+11x2+10x−6¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x3−7x2−17x−6
x2+11x+10x2+11x+10)x3−7x2−77x−70¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x3+0x2+60x+64
This gives us:
x3+4x2−7x−6x2+11x+10=(x−7)+60x+64x2+11x+10=(x−7)+4(15x+16)(x+1)(x+10)
