real analysis - Prove that $f(x) = x^ 2 + 1$ is continuous on R using a δ − ε proof.

For this problem, we weren't allowed to simply apply properties of continuous functions. My professor gave me some hints and tips but I'm still not sure how to finish out the proof. Here's what I have so far.

For ε, $|(x^2+1)-(a^2+1)|<ε$. He told me I want $|x-a|*|x+a|<ε$.

Then he said I should suppose δ=1 and $|x|=\leq 1+|a|$.

Next we did some work to end up with $|x-a|<\frac{ε}{2|a|+1}$.

And lastly, $δ=min(1,\frac{ε}{2|a|+1})$.

I've really been struggling with the δ and ε proofs and am not sure what to do next. Could someone work it through the rest of the way while explaining what they're doing? Thanks.