jmc

Let $p(x)=(a+x)(b+x)(c+x),$ which is a monic, third-degree polynomial in $x.$ Let $q(x)=p(x)-x,$ so $q(1)=q(2)=q(3)=0.$ Also, $q(x)$ is cubic and monic, so $$q(x)=(x-1)(x-2)(x-3).$$Hence, $p(x)=(x-1)(x-2)(x-3)+x.$ In particular, $p(4)=(3)(2)(1)+4=\boxed{10}.$