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Let's solve the given problem step by step. 

We are given the functions:

\[
f(x) = 2x^2
\]
\[
g(x) = x - 4
\]

### 1. Expression for \((g + f)(x)\):
\[
(g + f)(x) = g(x) + f(x) = (x - 4) + (2x^2) = 2x^2 + x - 4
\]

### 2. Expression for \((g - f)(x)\):
\[
(g - f)(x) = g(x) - f(x) = (x - 4) - (2x^2) = -2x^2 + x - 4
\]

### 3. Evaluation of \((g \cdot f)(-2)\):
To find \((g \cdot f)(-2)\), first we need to compute the product of \(g(x)\) and \(f(x)\) and then substitute \(x = -2\).

\[
(g \cdot f)(x) = g(x) \cdot f(x) = (x - 4) \cdot (2x^2)
\]

Expanding this expression:

\[
(g \cdot f)(x) = (x - 4)(2x^2) = 2x^3 - 8x^2
\]

Now, substitute \(x = -2\):

\[
(g \cdot f)(-2) = 2(-2)^3 - 8(-2)^2 = 2(-8) - 8(4) = -16 - 32 = -48
\]

### Final Answers:
- \((g + f)(x) = 2x^2 + x - 4\)
- \((g - f)(x) = -2x^2 + x - 4\)
- \((g \cdot f)(-2) = -48\)

Would you like any further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the derivative of \((g + f)(x)\)?
2. What is the value of \((g - f)(2)\)?
3. Can you express \((g \cdot f)(x)\) in factored form?
4. How would you solve for the roots of \((g + f)(x) = 0\)?
5. What happens to \((g \cdot f)(x)\) as \(x\) approaches infinity?

**Tip:** When multiplying polynomials, ensure to carefully distribute each term to avoid common errors.

Write the expressions for (g + f)(x) and (g - f)(x) and evaluate (g ⋅ f)(-2). The functions provided are f(x) = 2x^2 and g(x) = x - 4.

Learn how to solve the function operation problem involving polynomial functions, including (g + f)(x), (g - f)(x), and the evaluation of (g ⋅ f)(-2). This problem includes key algebraic concepts such as function addition, subtraction, and multiplication, suitable for Grades 9-12.

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