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Polynomial identities

Problem

Felix was asked whether the following equation is an identity:
left parenthesis, x, squared, plus, 1, right parenthesis, squared, equals, left parenthesis, x, squared, minus, 1, right parenthesis, squared, plus, left parenthesis, 2, x, right parenthesis, squared
He performed the following steps:
=(x2+1)2Step 1=x4+x2+x2+1Step 2=x4+2x2+1Step 3=x4+2x2+12x2+2x2Step 4=(x42x2+1)+4x2Step 5=(x21)2+(2x)2\begin{aligned} &\phantom{=}(x^2+1)^2 \\\\ \xhookrightarrow{\text{Step }1}\quad&=x^4+x^2+x^2+1 \\\\ \xhookrightarrow{\text{Step }2}\quad&=x^4+2x^2+1 \\\\ \xhookrightarrow{\text{Step }3}\quad&=x^4+2x^2+1-2x^2+2x^2 \\\\ \xhookrightarrow{\text{Step }4}\quad&=(x^4-2x^2+1)+4x^2 \\\\ \xhookrightarrow{\text{Step }5}\quad&=(x^2-1)^2+(2x)^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?
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