If at least one root of $2{\mathrm{x}}^2+3\mathrm{x}+5=0$ and ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0,\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{N}$  is common, then the maximum value of a+ b +c is

Roots of the equation $2{\mathrm{x}}^2+3\mathrm{x}+5=0$ are
$\mathrm{x}=\frac{-3\pm \sqrt{9-40}}{6}$ (imaginary roots) 
Hence, both roots coincide, so on comparing
$$\begin{gathered} \frac{\mathrm{a}}{2}=\frac{\mathrm{b}}{3}=\frac{\mathrm{c}}{5}=\mathrm{k} \\ \begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}=2\mathrm{k},\mathrm{b}=3\mathrm{k},\mathrm{c}=5\mathrm{k}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}+\mathrm{b}+\mathrm{c}=10\mathrm{k}\end{array} \end{gathered}$$
So, maximum value does not exist.