$ (x^4 + x^3 + 2x^2 + x + 1) + (x^4 - 1) > (2x^3 + x^2 + 2x + 1)\Rightarrow 2x^4 - x^3 + x^2 - x - 1 > 0$ $ (2x^3 + x^2 + 2x + 1) + (x^4 - 1) > (x^4 + x^3 + 2x^2 + x + 1)\Rightarrow x^3 - x^2 + x - 1 > 0$ $ (2x^3 + x^2 + 2x + 1) + (x^4 + x^3 + 2x^2 + x + 1) > (x^4 - 1)\Rightarrow 3x^3 + 3x^2 + 3x + 3 > 0$ These are all true for $ x > 1$ Or, one can realize that $ x^4+x^3+2x^2+x+1$ is the largest side, so only the second line is actually necessary.