A charge Q is uniformly distributed throughout the material of a...

The total charge $Q$ is uniformly distributed in the volume between $r_1$ and $r_2$.

The volume of the material of the hollow sphere is:
$V = \dfrac{4}{3}\pi(r_2^3 - r_1^3)$

The uniform volume charge density $\rho$ is:
$\rho = \dfrac{Q}{V} = \dfrac{Q}{\dfrac{4}{3}\pi(r_2^3 - r_1^3)}$

To find the electric field at a distance $x$ ($r_1 < x < r_2$), we consider a spherical Gaussian surface of radius $x$ concentric with the hollow sphere.

The charge enclosed by this Gaussian surface is the charge within the volume from $r_1$ to $x$:
$q_{enc} = \rho \times \dfrac{4}{3}\pi(x^3 - r_1^3)$

Substituting the value of $\rho$:
$q_{enc} = \dfrac{Q}{\dfrac{4}{3}\pi(r_2^3 - r_1^3)} \times \dfrac{4}{3}\pi(x^3 - r_1^3) = Q \left( \dfrac{x^3 - r_1^3}{r_2^3 - r_1^3} \right)$

According to Gauss's Law:
$\oint \vec{E} \cdot d\vec{A} = \dfrac{q_{enc}}{\varepsilon_0}$

Due to spherical symmetry, the electric field $E$ is constant in magnitude and points radially outward over the Gaussian surface:
$E (4\pi x^2) = \dfrac{Q(x^3 - r_1^3)}{\varepsilon_0(r_2^3 - r_1^3)}$

$E = \dfrac{Q(x^3 - r_1^3)}{4\pi\varepsilon_0 x^2 (r_2^3 - r_1^3)}$