The coefficient of $x^5$  in the expansion of ${\left(x^2-x-2\right)}^5$  is

$={\left(x\left(x-2\right)+1\left(x-2\right)\right)}^5$

$={\left(x-2\right)}^5{\left(x+1\right)}^5$

$={\left(1+x\right)}^5{\left(x-2\right)}^5$

$=\left[{}^5{C}_{0}1+{}^5{C}_{1}x+{}^5{C}_2{x}^2+{}^5{C}_3{x}^3+{}^5{C}_4{x}^4+{}^5{C}_5{x}^5\right]$

$\times \left[{}^5{C}_0.x^5-{}^5{C}_1{x}^4.2+{}^5{C}_2{x}^3{.2}^2-{}^5{C}_3{x}^2{2}^3+{}^5{C}_{4}x{.2}^4-{}^5{C}_5{2}^5\right]$

$=\left[{}^5{C}_0\times {}^5{C}_0+{}^5{C}_1\times \left(-{}^5{C}_1\left(2\right)\right)+{}^5{C}_2\times {}^5{C}_2\times 2^2+{}^5{C}_3\times \left(-{}^5{C}_3\right)2^3+{}^5{C}_4\times {}^5{C}_4{.2}^4+{}^5{C}_5\times -{}^5{C}_{5}2\right]$

$=1-50+400-800+400-32$

$=-81$