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.