1. The problem is to understand the shape of the graph of the function $$g(x) = (x + 4)^3 + 3$$. 2. This is a cubic function shifted horizontally and vertically. The base cubic function is $$f(x) = x^3$$, which has an S-shaped curve passing through the origin. 3. The term $$(x + 4)^3$$ shifts the graph 4 units to the left because replacing $$x$$ by $$x + 4$$ moves the graph left. 4. The $$+3$$ outside the cube shifts the graph 3 units up. 5. So the graph of $$g(x)$$ is the cubic curve shifted left by 4 and up by 3. 6. The graph will have an inflection point at $$x = -4$$, $$y = 3$$, where the curve changes concavity. 7. The graph looks like an S-shaped curve centered at $$(-4, 3)$$, increasing steeply for large positive and negative $$x$$ values. Final answer: The graph of $$g(x) = (x + 4)^3 + 3$$ is an S-shaped cubic curve shifted left 4 units and up 3 units, with an inflection point at $$(-4, 3)$$.