Factoring $x^4-2x^3+2x^2+x+4$ 
I need to show that the polynomial is not irreducible and I am trying to factor the polynomial 
  $$x^4-2x^3+2x^2+x+4$$

I checked from a calculator that it has a factor but how do I get it by myself?
I tried grouping but It didnt work I got 
$x^2(x^2-2x+2)+x+4$ And I dont know how should I proceed. My guts tell me that it should be of the form:
$(x^2-ax+k)(x^2+bx+l)$, should I just try to figure out the constants by trying out?

$$x^4-2x^3+2x^2+x+4$$
$$=(x^2+1)^2-2x^3+x+3$$
$$=(x^2+1)^2-2x(x^2+1)+3x+3$$
$$=(x^2+1)^2+x(x^2+1)-3x^3+3$$
$$=(x^2+1)^2+x(x^2+1)-3(x-1)(x^2+x+1)$$
$$=(x^2+x+1)(x^2+1-3x+3)$$
and you are done.
