Let's find the expression for $g(x+4)$ where $g(x) = x(x^2 - 4)$.

Step 1: Substitute $x+4$ into $g(x)$

We replace every $x$ in $g(x)$ with $x+4$: $g(x+4) = (x+4)((x+4)^2 - 4)$

Step 2: Simplify the expression

First, expand $(x+4)^2$: $(x+4)^2 = x^2 + 8x + 16$ Now subtract 4: $(x+4)^2 - 4 = (x^2 + 8x + 16) - 4 = x^2 + 8x + 12$

Step 3: Multiply the terms

Now multiply $(x+4)$ by $(x^2 + 8x + 12)$: $g(x+4) = (x+4)(x^2 + 8x + 12)$

Distribute $(x+4)$ across the terms inside the parentheses: $g(x+4) = (x+4)(x^2) + (x+4)(8x) + (x+4)(12)$ $g(x+4) = x^3 + 8x^2 + 12x + 4x^2 + 32x + 48$

Step 4: Combine like terms

$g(x+4) = x^3 + (8x^2 + 4x^2) + (12x + 32x) + 48$ $g(x+4) = x^3 + 12x^2 + 44x + 48$

Thus, the equation for $g(x+4)$ is: $g(x+4) = x^3 + 12x^2 + 44x + 48$

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the derivative of $g(x) = x(x^2 - 4)$?
2. How would the graph of $g(x+4)$ differ from $g(x)$?
3. What are the roots of the equation $g(x+4) = 0$?
4. How would you find $g'(x+4)$, the derivative of $g(x+4)$?
5. What is the value of $g(2)$ for both $g(x)$ and $g(x+4)$?

Tip: Always distribute and combine like terms carefully to avoid calculation mistakes.