To solve the problem, we need to simplify the given expression step by step. The expression involves fractions and polynomial multiplication, so let’s analyze each component:

Expression:

$\frac{3y^2 - 10y + 3}{3y^2 + 5y - 2} \cdot \frac{2y^2 - 3y - 20}{2y^2 - y - 15} \div (y - 4)$

Step 1: Factorize each polynomial

(a) $3y^2 - 10y + 3$

Factoring: $3y^2 - 10y + 3 = (3y - 1)(y - 3)$

(b) $3y^2 + 5y - 2$

Factoring: $3y^2 + 5y - 2 = (3y - 1)(y + 2)$

(c) $2y^2 - 3y - 20$

Factoring: $2y^2 - 3y - 20 = (2y + 5)(y - 4)$

(d) $2y^2 - y - 15$

Factoring: $2y^2 - y - 15 = (2y + 5)(y - 3)$

(e) $y - 4$

This term remains $y - 4$, as it cannot be further factored.

Step 2: Substitute the factored forms into the expression

$\frac{(3y - 1)(y - 3)}{(3y - 1)(y + 2)} \cdot \frac{(2y + 5)(y - 4)}{(2y + 5)(y - 3)} \div (y - 4)$

Step 3: Cancel common terms

• $3y - 1$ appears in the numerator and denominator, so cancel it.
• $2y + 5$ appears in the numerator and denominator, so cancel it.
• $y - 3$ appears in the numerator and denominator, so cancel it.

The simplified expression becomes: $\frac{y - 4}{y + 2} \cdot \frac{1}{y - 4}$

Step 4: Multiply and simplify

The $y - 4$ in the numerator and denominator cancels out: $\frac{1}{y + 2}$

Final Answer:

$\frac{1}{y + 2}$

Let me know if you need a detailed explanation of any step or further clarification. Here are some related questions to consider:

1. How do you determine the factors of a quadratic polynomial?
2. What happens if one term in the numerator or denominator has a common factor of zero?
3. Can we simplify the expression further if additional constraints are added to $y$?
4. How do we handle division of fractions in such expressions?
5. What are the domain restrictions for $y$ in this problem?

Tip: Always check for restrictions on $y$ to ensure no division by zero occurs in the expression. In this case, $y \neq -2, 4, 3, -5/2$.