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Let's break down and solve the given problem.

### Given Functions:
1. \( f(x) = 3x^3 - 2x^2 + 4x - 5 \)
2. \( g(x) = x^2 - 5x + 6 \)

### Problem:
We are asked to:
a) Compute \( f(x) \cdot g(x) \)  
b) Compute \( \frac{f(x)}{g(x)} \)

---

### Part (a) – Multiplying \( f(x) \) and \( g(x) \):
We will multiply \( f(x) \) and \( g(x) \) as polynomials.

\[
f(x) \cdot g(x) = (3x^3 - 2x^2 + 4x - 5) \cdot (x^2 - 5x + 6)
\]

We'll expand this expression step by step.

### Part (b) – Dividing \( f(x) \) by \( g(x) \):
We will divide \( f(x) \) by \( g(x) \) using polynomial long division.

\[
\frac{f(x)}{g(x)} = \frac{3x^3 - 2x^2 + 4x - 5}{x^2 - 5x + 6}
\]

I'll now calculate both parts.### Solutions:

#### Part (a) – \( f(x) \cdot g(x) \):
\[
f(x) \cdot g(x) = 3x^5 - 17x^4 + 32x^3 - 37x^2 + 49x - 30
\]

#### Part (b) – \( \frac{f(x)}{g(x)} \):
\[
\frac{f(x)}{g(x)} = \frac{3x^3 - 2x^2 + 4x - 5}{x^2 - 5x + 6}
\]

This is the simplified form of the division.

Would you like more details on any of the steps? 

Here are some related questions to further explore:

1. How do you perform polynomial long division?
2. How do you factor complex polynomials like \( f(x) \) or \( g(x) \)?
3. What are the roots of \( f(x) \) and \( g(x) \)?
4. How can you graph the resulting functions of \( f(x) \cdot g(x) \) and \( \frac{f(x)}{g(x)} \)?
5. What are the applications of polynomial multiplication and division in real-world problems?

**Tip:** Always check for common factors in polynomial division to simplify calculations easily.

Given f(x) = 3x^3 - 2x^2 + 4x - 5 and g(x) = x^2 - 5x + 6, find: a) f(x) * g(x) b) f(x) / g(x).

This problem involves multiplying and dividing two polynomials, f(x) = 3x^3 - 2x^2 + 4x - 5 and g(x) = x^2 - 5x + 6. Learn how to expand and simplify the products and perform polynomial long division. Suitable for Grades 9-12.

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