Let g(x)=x(x^2-4). Find equations for the following.     g(x+4)=

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.

Let g(x)=x(x^2-4). Find equations for the following. g(x+4)=

Discover how to find the equation for g(x+4) when given g(x) = x(x^2 - 4). This algebraic problem involves polynomial functions and function transformations, providing a detailed step-by-step solution. Suitable for high school students in Grades 9-12.

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