jmc

Adding $10x$ to both sides, we get $$x^5+5x^4+10x^3+10x^2+5x+1=10x+10.$$Then $(x+1{)}^5=10(x+1).$ Since $x+1\ne 0,$ we can divide both sides by $x+1,$ to get $(x+1{)}^4=\boxed{10}.$