Answer
See the detailed answer below.
Work Step by Step
The orange dashed circle in the figure below are the 3 Gaussian spheres.
According to Gauss's law,
$$\oint \vec Ed\vec A=\dfrac{Q_{in}}{\epsilon_0}$$
According to the uniform charge distribution in the shell, the shell has spherical symmetry. So the electric field is uniform.
$$EA=\dfrac{Q_{in}}{\epsilon_0}$$
Hence,
$$E=\dfrac{Q_{in}}{\epsilon_0A}$$
$$\color{blue}{\bf [a]}$$
when $r\gt R_{\rm out}$;
$$E=\dfrac{Q_{in}}{\epsilon_0A}=\dfrac{Q}{4\pi r^2\epsilon_0}$$
Hence,
$$\boxed{\vec E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}\;\hat r}\tag {Outward}$$
$$\color{blue}{\bf [b]}$$
when $r\lt R_{\rm in}$;
$$E=\dfrac{Q_{in}}{\epsilon_0A}=\dfrac{0}{4\pi r^2\epsilon_0}$$
Hence,
$$\boxed{\vec E=\bf 0\;\rm N/C}$$
$$\color{blue}{\bf [c]}$$
when $R_{\rm in}\lt r\lt R_{\rm out}$;
$$E=\dfrac{Q_{in}}{\epsilon_0A}$$
$$E=\dfrac{Q_{in}}{4\pi r^2\epsilon_0}\tag 1$$
where the volume charge density of the shell is given by
$$\rho_s=\dfrac{Q}{V}=\dfrac{Q}{\frac{4}{3}\pi (R_{\rm out}^3 -R_{\rm in}^3 )}=\dfrac{Q_{in}}{\frac{4}{3}\pi (r^3-R_{\rm in}^3)}$$
where $V=V_{\rm out}-V_{\rm in}$
Hence,
$$Q_{in}=\dfrac{\frac{4}{3}\pi (r^3-R_{\rm in}^3)Q }{\frac{4}{3}\pi(R_{\rm out}^3 -R_{\rm in}^3 ) }=\dfrac{(r^3-R_{\rm in}^3) Q }{ (R_{\rm out}^3 -R_{\rm in}^3 ) }$$
Plug into (1),
$$E=\dfrac{1}{4\pi r^2\epsilon_0}\dfrac{(r^3-R_{\rm in}^3) Q}{ (R_{\rm out}^3 -R_{\rm in}^3 ) }$$
$$\boxed{\vec E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q(r^3-R_{\rm in}^3)}{ r^2(R_{\rm out}^3 -R_{\rm in}^3 )}\;\hat r}\tag {Outward}$$
$$\color{blue}{\bf [d]}$$
when $r=R_{\rm in}$,
$$ E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q(R_{\rm in}^3-R_{\rm in}^3)}{ r^2(R_{\rm out}^3 -R_{\rm in}^3 )} =\bf 0\;\rm N/C$$
which is the same result of (b),
And when $r=R_{\rm out}$,
$$ E=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q(R_{\rm out}^3-R_{\rm in}^3)}{ r^2(R_{\rm out}^3 -R_{\rm in}^3 )} =\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{r^2}$$
which is the same result of (a),
So the previous boxed formula, in part (c), matches at both the inner and outer boundaries.
