Method for proving polynomial inequalities 
Let $x\in\mathbb{R}$. Prove that
  $\text{(a) }x^{10}-x^7+x^4-x^2+1>0\\
\text{(b) }x^4-x^2-3x+5>0$

Possibly it can be proved in a few different ways, but I have first tried to prove it reducing to a sum of squares. After too many attempts and using a trial-and-error method, I got
$$x^4\left({x^3-\frac12}\right)^2+\left({\frac12x^2-1}\right)^2+\frac12x^4>0$$
for $\text{(a)}$. My question is: is there any easier method to prove this for any polinomial which is always positive? Also, I am wondering if there is any other simplier method than reducing to sum of squares.

Just another way for $(a)$ is using the AM-GMs:
$$\frac12x^{10}+\frac12x^4 \ge x^7, \quad \frac12x^4+\frac12 \ge x^2$$
$$\implies x^{10}-x^7+x^4-x^2+1 \ge \frac12+\frac12x^{10}>0$$
and similarly for $(b)$:
$$\frac12x^4+\frac12 \ge x^2, \quad \frac12x^4+\frac32+\frac32+\frac32 \ge 2\times 3^{3/4}x> 3x$$
Not always applicable, of course.  
Also you may want to know that for univariate polynomials theoretically you can always express as a sum of squares, and while tedious, somehow the end result is more satisfying...
