= Solution
The displayed <Rodrigues' formula> gives the <Legendre polynomial>
$$
q_4(x)=\frac18(35x^4-30x^2+3).
$$
A four-node Gaussian rule is exact through degree $2\cdot4-1=7$. Solving $q_4(x)=0$ first for $x^2$ gives the four nodes
$$
\boxed{c_k\in
\left\{
-\sqrt{\frac{15+2\sqrt{30}}{35}},
-\sqrt{\frac{15-2\sqrt{30}}{35}},
\sqrt{\frac{15-2\sqrt{30}}{35}},
\sqrt{\frac{15+2\sqrt{30}}{35}}
\right\}}.
$$
The rule is not exact for every polynomial of degree eight: its quadrature value on $q_4^2$ is zero because every node is a root of $q_4$, whereas
$$
\boxed{\int_{-1}^1q_4(x)^2\,dx>0.}
$$
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