\(I=3x^2+4xy+4y^2+5x=\left(2x^2+5x+\dfrac{25}{8}\right)+\left(x^2+4xy+4y^2\right)-\dfrac{25}{8}=\left(\sqrt{2}x+\dfrac{5\sqrt{2}}{4}\right)^2+\left(x+2y\right)^2-\dfrac{25}{8}\ge-\dfrac{25}{8}\)
\(minI=-\dfrac{25}{8}\Leftrightarrow\)\(\left\{{}\begin{matrix}x=-\dfrac{5}{4}\\y=\dfrac{5}{8}\end{matrix}\right.\)