4. The conic section $(4 x-y+3)^{2}=9 x^{2}+10$ has an eccentricity of $\qquad$ .
4. $\sqrt{10}$.

The equation of the conic section is
$$
(x-y+3)(7 x-y+3)=10 \text {, }
$$

which is a hyperbola with asymptotes $l_{1}: x-y+3=0$ and $l_{2}: 7 x-y+3=0$.
As shown in Figure 2.
Let the angle from $l_{2}$ to $l_{1}$ be $\theta$. Then $\theta \in(0, \pi)$, and
$$
\begin{array}{l}
\tan \theta=\frac{k_{1}-k_{2}}{1+k_{1} k_{2}}=\frac{1-7}{1+1 \times 7}=-\frac{3}{4} \\
\Rightarrow \cos \theta=-\frac{4}{5} . \\
\text { Hence } e=\sqrt{\frac{c^{2}}{a^{2}}}=\sqrt{1+\frac{b^{2}}{a^{2}}}=\sqrt{1+\tan ^{2} \frac{\theta}{2}} \\
=\sqrt{\frac{1}{\cos ^{2} \frac{\theta}{2}}}=\sqrt{\frac{2}{1+\cos \theta}}=\sqrt{10} .
\end{array}
$$