Answer
$(x+1)(x+1)(x^2-x+1)$
Work Step by Step
We factor by grouping as follows
$$
x^4 + x^3 + x+ 1= x^3(x+1)+(x+1)
.$$
Now, factoring $x+1$ out, we have
$$
x^3(x+1)+(x+1) =(x+1)(x^3+1)
.$$
Since $(x^3 +1)$ is a sum of two cubes we have
$$
(x+1)(x^3+1)=(x+1)(x+1)(x^2-x+1)
.$$
where we used the formula $(x+y)^3=(x+y)(x^2-xy+y^2)$.
