$2{{x}^{2}}+(c+2)x+8=0\left\{ \begin{matrix}s=\alpha +\beta =-\frac{b}{a}=\frac{-(c+2)}{2}  \\p=\alpha \beta =\frac{c}{a}=\frac{8}{2}=4  \\\end{matrix} \right.$ $\Rightarrow \left\{ \begin{matrix}{{t}_{1}}=\sqrt{\alpha \beta }  \\{{t}_{2}}=2\sqrt{\alpha \beta }  \\\end{matrix} \right.$ $S={{t}_{1}}+{{t}_{2}}=\sqrt{\alpha \beta }+2\sqrt{\alpha \beta }$ $\begin{align}& =3\sqrt{\alpha \beta }=3\sqrt{p}=3\sqrt{4}=6 \\ & p={{t}_{1}}{{t}_{2}}=\sqrt{\alpha \beta }(2\sqrt{\alpha \beta })=2\alpha \beta =2p=2\times (4)=8 \\ \end{align}$ $\left. \begin{matrix}\Rightarrow {{x}^{2}}-Sx+p=0\Rightarrow {{x}^{2}}-6x+8=0  \\{{x}^{2}}+bx+c=0  \\\end{matrix} \right\}\Rightarrow c=8$ $2{{x}^{2}}+(c+2)x+8=0\xrightarrow{c=8}2{{x}^{2}}+10x+8=0$ $\Rightarrow \alpha +\beta =\frac{-b}{a}=\frac{-10}{2}=-5$