The task in the image is to graph the function $f(x) = x^3 + 4x^2 - 5$ by using a table of values.

Steps:

1. Create a table of values by choosing some $x$-values and calculating the corresponding $f(x)$-values. Let's choose $x = -4, -2, 0, 2, 4$ based on the grid provided.

2. Calculate the values of the function: $f(x) = x^3 + 4x^2 - 5$ For each $x$, substitute the value into the function:

   • When $x = -4$: $f(-4) = (-4)^3 + 4(-4)^2 - 5 = -64 + 64 - 5 = -5$
   • When $x = -2$: $f(-2) = (-2)^3 + 4(-2)^2 - 5 = -8 + 16 - 5 = 3$
   • When $x = 0$: $f(0) = (0)^3 + 4(0)^2 - 5 = -5$
   • When $x = 2$: $f(2) = (2)^3 + 4(2)^2 - 5 = 8 + 16 - 5 = 19$
   • When $x = 4$: $f(4) = (4)^3 + 4(4)^2 - 5 = 64 + 64 - 5 = 123$

3. Plot the points on the graph: The points you can plot are: $(-4, -5), (-2, 3), (0, -5), (2, 19), (4, 123)$

4. Sketch the graph: After plotting these points, you will draw a smooth curve passing through them, noting that this is a cubic function. The shape of a cubic function usually includes a turning point, and it goes to negative infinity as $x \to -\infty$ and positive infinity as $x \to +\infty$.

Would you like me to generate the graph for you or explain further details about this? Here are some follow-up questions that can deepen your understanding:

1. How do cubic functions typically behave for large positive and negative values of $x$?
2. Why do we include turning points when sketching cubic functions?
3. What are the general characteristics of the graph of a polynomial of degree 3?
4. How would you determine the turning points or local minima/maxima of this cubic function?
5. What is the significance of the leading term $x^3$ in determining the graph's overall shape?

Tip: For polynomial functions, the degree of the leading term gives important information about the general shape of the graph.