Solve the equation. \[ \log _{3}\left(x^{2}+1\right)=4 \]

Step 1 ：Convert the logarithmic equation into an exponential equation. The base of the logarithm becomes the base of the power, the right hand side of the equation becomes the exponent and the argument of the logarithm becomes the result. So, the equation becomes \(3^{4} = x^{2} + 1\).

Step 2 ：Simplify the equation to get \(81 = x^{2} + 1\).

Step 3 ：Subtract 1 from both sides to get \(x^{2} = 80\).

Step 4 ：Take the square root of both sides to get \(x = \pm \sqrt{80}\).

Step 5 ：Simplify \(\sqrt{80}\) to get \(x = \pm 4\sqrt{5}\).

Step 6 ：Final Answer: The solutions to the equation are \(\boxed{-4\sqrt{5}}\) and \(\boxed{4\sqrt{5}}\).