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

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

$=-\left[{}^5\mathrm{C}_0{2}^5-{}^5\mathrm{C}_1{2}^4\mathrm{x}+{}^5\mathrm{C}_2{2}^3{\mathrm{x}}^2+\dots \dots \right]\left[{}^5\mathrm{C}_0+{}^5\mathrm{C}_1\mathrm{x}+{}^5\mathrm{C}_2{\mathrm{x}}^2+\dots \dots \right]$

$\therefore$Coefficient of ${\mathrm{x}}^2$ is $=-{}^5\mathrm{C}_0{2}^5·{}^5\mathrm{C}_2+{}^5\mathrm{C}_1·2^4·{}^5\mathrm{C}_1-{}^5\mathrm{C}_2{2}^3=0.$