Answer
The function has a maximum value of $-6$.
Work Step by Step
The given function is
$\Rightarrow f(x)=-3x^2-6x-9$
Add $6$ to each side.
$\Rightarrow f(x)+6=-3x^2-6x-9+6$
Simplify.
$\Rightarrow f(x)+6=-3x^2-6x-3$
Factor out $-3$ from the right side.
$\Rightarrow f(x)+6=-3(x^2+2x+1)$
Write the right side as the square of a binomial.
$\Rightarrow f(x)+6=-3(x+1)^2$
Write in vertex form.
$\Rightarrow f(x)=-3(x+1)^2-6$
The vertex is $(-1,-6)$. Because $a$ is negative $(a=-3)$, the parabola opens down and the $y-$coordinate of the vertex is the maximum value.
Hence, the function has a maximum value of $-6$.
