Factor out the $3$ and separate it from the constant.

Complete the square.
Add the value that completes the square inside the parentheses
and subtract it outside.
(Why is the value subtracted $3$ not $1$?)

$y = 3x^{2} + 6x + 1$
$y = 3\left(x^{2} + 2x\right) + 1$

$y = 3\left(x^{2} + 2x + 1\right) + 1 − 3$

$y = 3\left(x + 1\right)^{2} − 2$