Answer
$(x+1)^2(x^2-x+1)$
Work Step by Step
Group the first two terms together and group the last two terms together.
$x^4+x^3+x+1=(x^4+x^3)+(x+1)$
Factor out the GCF in each group.
$=x^3(x+1)+1(x+1)$
Factor out $(x+1)$.
$=(x+1)(x^3+1)$
$=(x+1)(x^3+1^3)$
Use the special formula $a^3+b^3=(a+b)(a^2-ab+b^2)$ where $a=x$ and $b=1$.
$=(x+1)[(x+1)(x^2-x(1)+1^2)]$
$=(x+1)^2(x^2-x+1)$
Hence, the completely factored form of the given expression is $(x+1)^2(x^2-x+1)$.
