$$\begin{gathered} \mathrm{If} \mathrm{a}, \mathrm{b}, \mathrm{c} \mathrm{are} \mathrm{non}-\mathrm{zero} \mathrm{real} \mathrm{numbers} \mathrm{such} \mathrm{that} \\ {\int }_0^1\left(1+{\cos }^8 \mathrm{x}\right)\left({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}\right) \mathrm{dx}= \\ {\int }_0^2\left(1+{\cos }^8 \mathrm{x}\right)\left({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}\right) \mathrm{dx}= \\ \mathrm{then} \mathrm{the} \mathrm{equation} {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0 \mathrm{will} \mathrm{have} \end{gathered}$$

$\mathrm{both} \mathrm{the} \mathrm{roots} \mathrm{between} 0 \mathrm{and} 1$

$\mathrm{both} \mathrm{the} \mathrm{roots} \mathrm{between} 1 \mathrm{and} 2$

$\mathrm{both} \mathrm{the} \mathrm{roots} \mathrm{between} \left(3, \infty \right)$

$$\begin{gathered} \mathrm{one} \mathrm{root} \mathrm{between} 0 \mathrm{and} 1 \mathrm{and} \mathrm{other} \mathrm{root} \\ \mathrm{between} 1\mathrm{and} 2 \end{gathered}$$