How to notice that $3^2 + (6t)^2 + (6t^2)^2$ is a binomial expansion. The other day during a seminar, in a calculation, a fellow student encountered this expression:
$$\sqrt{3^2 + (6t)^2 + (6t^2)^2}$$
He, without much thinking, immediately wrote down:
$$(6t^2+3)$$
What bothers me, is that I didn't see that. Although I suspected there might be some binomial expansion, I was too confused by the middle term to see it. Even after actually seeing the answer, it took me a good deal to realize what's happening.
My initial idea, by the way, was to substitute for $t^2$ and solve a quadratic equation. Horribly slow.
My questions is perhaps tackling a greater problem. I now see what happened, I can "fully" understand binomial expansions, yet I feel like I might end up encountering a similar expression and yet again not be able to see the result. My questions is thus:
How can I properly practice so I don't repeat this mistake?
This is perhaps too much of a soft-question, but I feel like many others have been in a position of understanding a problem, yet not feeling like they would be able to replicate it, and might actually have an answer that goes well beyond the obvious "just practice more".
Thanks for any input.

$\bigcirc^2+\triangle+\square^2$ is a complete square if $\left|\triangle\right|=2\cdot\bigcirc\cdot\triangle$.
For example, $49x^2-42x+9$ is a complete square since the only candidate for $\triangle$ in $42x$, and indeed $42x=2\cdot7x\cdot3$.  In our special case we have three candidates for $\triangle$; your fellow student had three candidates for $\triangle$. “Without much thinking” -- a bit of thinking was implied, since the three choices -- he verified that $(6t)^2=2\cdot3\cdot6t^2$.
