[tex]\displaystyle \\
\text{Se cere sa aratam ca: } ~E(x)=x^2+x+1 \ \textgreater \ 0~~\text{pentru oricare }~x \in R. \\ \\
\text{Rezolvare: } \\ \\
x = 2\cdot x\cdot \frac{1}{2} \\ \\
1 = \frac{1}{4} +\frac{3}{4} \\ \\
E(x) = x^2+x+1 = x^2 + 2\cdot x\cdot \frac{1}{2} +\frac{1}{4} +\frac{3}{4}= \\ \\
= \underbrace{x^2 + 2\cdot x\cdot \frac{1}{2} +\left(\frac{1}{2} \right)^2}_{\text{patratul unui binom}} +\frac{3}{4}= \boxed{\left(x + \frac{1}{2} \right)^2 + \frac{3}{4}}[/tex]
[tex]\displaystyle \\\left(x + \frac{1}{2}\right)^2 \geq 0 ~~~\text{ pentru ca este patratul unei expresii.} \\ \\
\frac{3}{4} \ \textgreater \ 0~~~ \text{deoarece este un numar pozitiv} \\ \\ \\
\Longrightarrow~~~\boxed{\left[ \left(x + \frac{1}{2} \right)^2 + \frac{3}{4} \right] \ \textgreater \ 0}[/tex]
[tex]\Longrightarrow ~~~\boxed{E(x)=x^2+x+1 \ \textgreater \ 0~~\text{pentru oricare }~x \in R.}[/tex]
