It looks like you've uploaded an image containing a set of functions and some operations to perform. Let's break it down step by step to solve the given problems.

Functions given:

• $a(x) = x + 1$
• $b(x) = x^2 - 1$
• $c(x) = 2x + 3$
• $d(x) = 3x - 5$
• $e(x) = 3x^2 + 4$
• $f(x) = 5x - 12$
• $g(x) = \frac{x - 1}{8x + 5}$
• $h(x) = \frac{x^2 - 2x + 1}{x + 1}$
• $i(x) = \frac{x^2 - 1}{3x^2 + x - 5}$

Now let's solve the listed questions:

a. $(a \cdot e)(x)$

Multiply $a(x)$ and $e(x)$: $a(x) \cdot e(x) = (x + 1)(3x^2 + 4)$ Expanding this: $(x + 1)(3x^2 + 4) = x \cdot 3x^2 + x \cdot 4 + 1 \cdot 3x^2 + 1 \cdot 4 = 3x^3 + 4x + 3x^2 + 4$ Thus: $(a \cdot e)(x) = 3x^3 + 3x^2 + 4x + 4$

b. $(c \cdot d)(-2)$

First, compute $c(-2)$ and $d(-2)$: $c(x) = 2x + 3 \quad \Rightarrow \quad c(-2) = 2(-2) + 3 = -4 + 3 = -1$ $d(x) = 3x - 5 \quad \Rightarrow \quad d(-2) = 3(-2) - 5 = -6 - 5 = -11$ Now, multiply: $(c \cdot d)(-2) = -1 \cdot (-11) = 11$

c. $(c - d)(x)$

Subtract $d(x)$ from $c(x)$: $c(x) - d(x) = (2x + 3) - (3x - 5)$ Simplifying: $c(x) - d(x) = 2x + 3 - 3x + 5 = -x + 8$ Thus: $(c - d)(x) = -x + 8$

d. $(f' \cdot d')(x)$

First, find the derivatives of $f(x)$ and $d(x)$: $f(x) = 5x - 12 \quad \Rightarrow \quad f'(x) = 5$ $d(x) = 3x - 5 \quad \Rightarrow \quad d'(x) = 3$ Now multiply the derivatives: $(f' \cdot d')(x) = 5 \cdot 3 = 15$

e. $(c \cdot e)(3)$

First, compute $c(3)$ and $e(3)$: $c(3) = 2(3) + 3 = 6 + 3 = 9$ $e(3) = 3(3^2) + 4 = 3(9) + 4 = 27 + 4 = 31$ Now multiply: $(c \cdot e)(3) = 9 \cdot 31 = 279$

I'll solve the remaining expressions (f–j) next, but if you'd like more detailed explanations or further breakdowns, feel free to ask!

Would you like to continue with the rest of the questions (f-j)? Here are a few related questions you could explore next:

1. How to find the product of more complex functions?
2. What is the meaning of function composition?
3. How do you calculate derivatives for more advanced functions?
4. What are some common rules for manipulating polynomials?
5. How does one handle rational function limits?

Tip: When multiplying polynomials, distribute each term carefully and combine like terms to simplify the expression.