1. The problem asks us to find the formula for the function $g(x)$ which is a transformation of $f(x) = |x|$ shifted left 3 units and down 5 units. 2. The general rule for horizontal shifts is: if $f(x)$ is shifted left by $h$ units, the new function is $f(x + h)$. 3. The rule for vertical shifts is: if $f(x)$ is shifted down by $k$ units, the new function is $f(x) - k$. 4. Applying these rules to $f(x) = |x|$, shifting left 3 units means replacing $x$ with $x + 3$, so we get $|x + 3|$. 5. Shifting down 5 units means subtracting 5 from the function, so we get $|x + 3| - 5$. 6. Therefore, the formula for $g(x)$ is: $$g(x) = |x + 3| - 5$$ 7. This matches the description of the graph with vertex at $(-3, -5)$ and arms passing through $(-4, -4)$ and $(-2, -4)$. Final answer: $$g(x) = |x + 3| - 5$$