Answer
$-3(x+2)^2-5$
See graph.
Work Step by Step
Step 1. $f(x)=-3x^2-12x-17=-3(x^2+4x+4)+12-17=-3(x+2)^2-5$
Step 2. To graph $f(x)$, start with $y=x^2$, shift $2$ unit to the left, stretch vertically by a factor of 3, reflect across the x-axis, then shift $5$ units down. See graph.