f(x) = |x + 3| is translated 5 units down, meaning this is a vertical translation (up and down). Vertical translations are written outside of the "parentheses" (could be sq. root, abs. value symbols) while horizontal translations are written inside the parentheses.
Since this is a vertical translation the 5 units down would be written outside of the abs. value symbols. It would be a negative 5 because in vertical translations, up = + and down = -.
Your final answer is g(x) = |x + 3| - 5.