Factor the following four term polynomial

x^3+4x^2+x+4

All you have to do here is group:

(x³+4x²) + (x+4)

Then, GCF the first group (second one is already in its simplest form):

x²(x+4) + (x+4)

Last, complete (simplify):

(x+4)(x²+1)

Use grouping

(x^3+4x^2)+(x+4)

Factor the GCF from each parenthesis

x^2(x+4)+1(x+4)

Factor (x+4)

(x+4)(x^2+1)

x^3+4x^2+x+4 = x^2(x+4) + (x+4) = (x+4)(x^2+1) Over real coefficients
You may continue over complex value of coefficients to
(x+4)(x+i)(x- i)

x^3 + 4x^2 + x + 4 =
x^2(x + 4) + 1(x + 4) =
(x^2 + 1)(x + 4)

x^3+x+4x^2+4
=x(x^2+1)+4(x^2+1)
=(x+4)(x^2+1).