How can I produce $\sqrt{4+4t^2+t^4}=2+t^2$ $\sqrt{4+4t^2+t^4}=2+t^2$

That is the solution my calculator produces but I don't understand how it gets that solution.

If I factor it I get:

$\sqrt{4(1+t^2)+t^4} = 2\sqrt{1+t^2+t^4}$

$2\sqrt{1+t^2(1+t^2)}$

So how do I get from there to $2+t^2$
Since $(2+t^2)^2=4+4t^2+t^4$ then you can deduce your relation.
You can factor it a different way. Factor the entire thing, not just the first two terms.