$\begin{array}{l} {x^4} + \sqrt {{x^5}} + x - \sqrt x + 1\\ = {x^4} + {x^2}\sqrt x + x - \sqrt x + 1\\ = {\left( {{x^2} + \dfrac{1}{2}\sqrt x } \right)^2} + \dfrac{3}{4}x - \sqrt x + 1\\ = {\left( {{x^2} + \dfrac{1}{2}\sqrt x } \right)^2} + \dfrac{3}{4}\left( {x - \dfrac{4}{3}\sqrt x + \dfrac{4}{9}} \right) + \dfrac{2}{3}\\ = {\left( {{x^2} + \dfrac{1}{2}\sqrt x } \right)^2} + \dfrac{3}{4}{\left( {\sqrt x - \dfrac{2}{3}} \right)^2} + \dfrac{2}{3} >\dfrac{2}{3}>0 \end{array}$