$4 \cdot 78$ Solve the equation $3^{x^{2}+1}+3^{x^{2}-1}-270=0$.
［Solution］The original equation is
$$
\begin{aligned}
3 \cdot 3^{x^{2}}+\frac{1}{3} \cdot 3^{x^{2}}-270 & =0, \\
10 \cdot 3^{x^{2}} & =810, \\
3^{x^{2}} & =3^{4}, \\
x^{2} & =4, \\
x & = \pm 2 .
\end{aligned}
$$

Upon verification, $x= \pm 2$ are solutions to the original equation.