$M=5{{a}^{2}}+7{{b}^{2}}-6ab-4a-8b+2023$
$M=\left( 3{{a}^{2}}-6ab+3{{b}^{2}} \right)+\left( 2{{a}^{2}}-4a+2 \right)+\left( 4{{b}^{2}}-8b+4 \right)+2017$
$M=2\left( {{a}^{2}}-2ab+{{b}^{2}} \right)+2\left( {{a}^{2}}-2a+1 \right)+4\left( {{b}^{2}}-2b+1 \right)+2017$
$M=2{{\left( a-b \right)}^{2}}+2{{\left( a-1 \right)}^{2}}+4{{\left( b-1 \right)}^{2}}+2017$
Vì ${{\left( a-b \right)}^{2}}\ge 0\forall a;b$; ${{\left( a-1 \right)}^{2}}\ge 0\forall a$; ${{\left( b-1 \right)}^{2}}\ge 0\forall b$
Nên $2{{\left( a-b \right)}^{2}}+2{{\left( a-1 \right)}^{2}}+4{{\left( b-1 \right)}^{2}}+2017\ge 2017\forall a;b$
Dấu “=” xảy ra khi $\left\{ \begin{align}
& {{\left( a-b \right)}^{2}}=0 \\
& {{\left( a-1 \right)}^{2}}=0 \\
& {{\left( b-1 \right)}^{2}}=0 \\
\end{align} \right.\Leftrightarrow a=b=1$
Vậy GTNN của M là 2017 khi a = b = 1