Answer is D. The equations given in the question imply two things:
$\int_{0}^{1} f(x) dx = 0 \\ \int_{1}^{2} f(x) dx = 0$
by using properties of definite integrals
where $f(x) = (1+ \cos ^8 x)(ax^2+bx+c)$
Let us consider the first equation. We know that definite integration is the area under the curve with area above x axis as positive and below x axis as negative. When you are getting the integration as $0$, that means curve cuts the x - axis atleast once in the given interval . but
$1 + \cos ^8 x > 0$ as $1>0$ and $\cos ^8 x \geq 0$ (even power is always nonegative)
That means quadratic equation has atleast one root in each of the intervals $(0,1)$ and $(1,2)$ .
But it is a quadratic equation which cannot have more than two distinct real roots .
So $D$ is correct.