Let a, b, c be non zero real numbers such that ${\int }_0^1 \left(1+{\cos }^8\mathrm{x}\right)\left({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}\right)\mathrm{dx}={\int }_0^2 \left(1+{\cos }^8\mathrm{x}\right)\left({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}\right)\mathrm{dx}$ Then the quadratic equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ has 

$$\begin{gathered} \text{ Let }\mathrm{f}\left(\mathrm{x}\right)={\int }_0^x\left(1+{\cos }^8\mathrm{t}\right)\left(a{t}^2+bt+\mathrm{c}\right)dt \\ {\mathrm{f}}^1\left(\mathrm{x}\right)=\left(1+{\cos }^8\mathrm{x}\right)\left({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}\right) \end{gathered}$$
From given condition $\mathrm{f}\left(0\right)=\mathrm{f}\left(1\right);\mathrm{f}\left(1\right)=\mathrm{f}\left(2\right)$

By Rolles theorem for $\mathrm{f}\left(\mathrm{x}\right)\text{ in }[0,1]{\mathrm{f}}^1\left(\mathrm{\alpha }\right)=0,\text{ at least }$

one $\alpha$ such that, $0<\alpha <1$ 

By Rolles theorem for $\mathrm{f}\left(\mathrm{x}\right)\text{ in }[1,2]{\mathrm{f}}^1\left(\mathrm{\beta }\right)=0$ at least
$\text{ one }\mathrm{\beta }\text{ such that }1<\mathrm{\beta }<2$

$$\begin{gathered} {\text{f}}^1\left(\mathrm{\beta }\right)=0\Rightarrow \left(1+{\cos }^8\mathrm{\beta }\right)\left(a{\beta }^2+b\beta +\mathrm{c}\right)=0 \\ \Rightarrow {\mathrm{a\beta }}^2+\mathrm{b\beta }+\mathrm{c}=0 \\ \beta is root of a{x}^2+bx+c=0 \end{gathered}$$