Polynomial identities

Felix was asked whether the following equation is an identity:

(x^{2} + 1)^{2} = (x^{2} - 1)^{2} + (2 x)^{2}

He performed the following steps:

\begin{aligned} (x^{2} + 1)^{2} \\ \overset{Step 1}{↪} & = x^{4} + x^{2} + x^{2} + 1 \\ \overset{Step 2}{↪} & = x^{4} + 2 x^{2} + 1 \\ \overset{Step 3}{↪} & = x^{4} + 2 x^{2} + 1 - 2 x^{2} + 2 x^{2} \\ \overset{Step 4}{↪} & = (x^{4} - 2 x^{2} + 1) + 4 x^{2} \\ \overset{Step 5}{↪} & = (x^{2} - 1)^{2} + (2 x)^{2} \end{aligned}

For this reason, Felix stated that the equation is a true identity.

Is Felix correct? If not, in which step did he make a mistake?

Algebra 2