\(\sqrt{x^2-\frac{1}{4}+\sqrt{x^2+x+\frac{1}{4}}}=\frac{1}{2}\left(2x^3+x^2+2x+1\right)\)
\(\Leftrightarrow\sqrt{x^2-\frac{1}{4}+\sqrt{\left(x+\frac{1}{2}\right)^2}}=\frac{1}{2}\left(2x+1\right)\left(x^2+1\right)\)
\(\Leftrightarrow\sqrt{x^2-\frac{1}{4}+x+\frac{1}{2}}=\left(x+\frac{1}{2}\right)\left(x^2+1\right)\)
\(\Leftrightarrow\sqrt{\left(x+\frac{1}{2}\right)^2}=\left(x+\frac{1}{2}\right)\left(x^2+1\right)\)
\(\Leftrightarrow x^2+1=\frac{x+\frac{1}{2}}{x+\frac{1}{2}}=1\)
\(\Leftrightarrow x=0\)
