Answer
$3(x+1)^2-2$
See graph.
Work Step by Step
Step 1. $f(x)=3x^2+6x+1=3(x^2+2x+1)-3+1=3(x+1)^2-2$
Step 2. To graph $f(x)$, start with $y=x^2$, shift $1$ unit to the left, stretch vertically by a factor of 3, then shift $2$ units down. See graph.