Let's solve each part of the problem based on the given functions:

Given:

• $f(x) = 3x + 2x^2 - 10$
• $g(x) = 5x - x + 15^2$

First, simplify $g(x)$:

• $g(x) = 5x - x + 15^2 = 4x + 225$

A. $(f + g)(x)$

We add $f(x)$ and $g(x)$: $(f + g)(x) = f(x) + g(x) = (3x + 2x^2 - 10) + (4x + 225)$ $= 2x^2 + (3x + 4x) + (-10 + 225) = 2x^2 + 7x + 215$

B. $(f + g)(5)$

Substitute $x = 5$ into $(f + g)(x)$: $(f + g)(5) = 2(5)^2 + 7(5) + 215 = 2(25) + 35 + 215 = 50 + 35 + 215 = 300$

C. $(f - g)(x)$

We subtract $g(x)$ from $f(x)$: $(f - g)(x) = f(x) - g(x) = (3x + 2x^2 - 10) - (4x + 225)$ $= 2x^2 + (3x - 4x) + (-10 - 225) = 2x^2 - x - 235$

D. $(f - g)(-6)$

Substitute $x = -6$ into $(f - g)(x)$: $(f - g)(-6) = 2(-6)^2 - (-6) - 235 = 2(36) + 6 - 235 = 72 + 6 - 235 = -157$

E. $(f \cdot g)(x)$

Multiply $f(x)$ by $g(x)$: $(f \cdot g)(x) = (3x + 2x^2 - 10)(4x + 225)$ Expanding this: $= 3x(4x + 225) + 2x^2(4x + 225) - 10(4x + 225)$ $= 12x^2 + 675x + 8x^3 + 450x^2 - 40x - 2250$ $= 8x^3 + (12x^2 + 450x^2) + (675x - 40x) - 2250$ $= 8x^3 + 462x^2 + 635x - 2250$

F. $(f \cdot g)(2)$

Substitute $x = 2$ into $(f \cdot g)(x)$: $(f \cdot g)(2) = 8(2)^3 + 462(2)^2 + 635(2) - 2250$ $= 8(8) + 462(4) + 635(2) - 2250 = 64 + 1848 + 1270 - 2250$ $= 3182 - 2250 = 932$

G. $(f / g)(x)$

Divide $f(x)$ by $g(x)$: $(f / g)(x) = \frac{f(x)}{g(x)} = \frac{3x + 2x^2 - 10}{4x + 225}$

H. $(f / g)(3)$

Substitute $x = 3$ into $(f / g)(x)$: $f(3) = 3(3) + 2(3)^2 - 10 = 9 + 18 - 10 = 17$ $g(3) = 4(3) + 225 = 12 + 225 = 237$ $(f / g)(3) = \frac{17}{237}$

Final Answers:

• A. $(f + g)(x) = 2x^2 + 7x + 215$
• B. $(f + g)(5) = 300$
• C. $(f - g)(x) = 2x^2 - x - 235$
• D. $(f - g)(-6) = -157$
• E. $(f \cdot g)(x) = 8x^3 + 462x^2 + 635x - 2250$
• F. $(f \cdot g)(2) = 932$
• G. $(f / g)(x) = \frac{3x + 2x^2 - 10}{4x + 225}$
• H. $(f / g)(3) = \frac{17}{237}$

Would you like any further explanation on these steps?

Here are five questions you can explore further:

1. How do you perform long division on rational functions?
2. What are the properties of polynomials when added or subtracted?
3. How can factoring help in simplifying products of functions?
4. What are the practical applications of multiplying and dividing functions?
5. How does one find the domain of a function involving division?

Tip: When multiplying polynomials, remember to distribute each term in one polynomial to every term in the other.