Let's solve the matching of multiplication problems with their simplified polynomial results.

Left Column (Multiplication Problems):

1. $2x(4x^2 + 3x - 1)$
2. $(2x - 3)(4x + 5)$
3. $4x(2x)$
4. $(2x + 3)(4x^2 + 2x - 5)$

Right Column (Simplified Polynomials):

• $8x^2$
• $8x^2 + 2$
• $8x^2 - 2x - 15$
• $8x^3 + 6x^2 - 2x$
• $8x^3 + 6x^2 - 2x - 15$
• $8x^3 + 16x^2 - 4x - 15$

Step-by-step Solutions:

1. For $2x(4x^2 + 3x - 1)$: $2x \cdot 4x^2 + 2x \cdot 3x - 2x \cdot 1 = 8x^3 + 6x^2 - 2x$ Match: $8x^3 + 6x^2 - 2x$

2. For $(2x - 3)(4x + 5)$: Expanding using the distributive property (FOIL): $(2x)(4x) + (2x)(5) - (3)(4x) - (3)(5) = 8x^2 + 10x - 12x - 15 = 8x^2 - 2x - 15$ Match: $8x^2 - 2x - 15$

3. For $4x(2x)$: $4x \cdot 2x = 8x^2$ Match: $8x^2$

4. For $(2x + 3)(4x^2 + 2x - 5)$: Expanding using distributive property: $(2x)(4x^2) + (2x)(2x) + (2x)(-5) + (3)(4x^2) + (3)(2x) + (3)(-5)$ $= 8x^3 + 4x^2 - 10x + 12x^2 + 6x - 15 = 8x^3 + 16x^2 - 4x - 15$ Match: $8x^3 + 16x^2 - 4x - 15$

Final Matches:

1. $2x(4x^2 + 3x - 1)$ → $8x^3 + 6x^2 - 2x$
2. $(2x - 3)(4x + 5)$ → $8x^2 - 2x - 15$
3. $4x(2x)$ → $8x^2$
4. $(2x + 3)(4x^2 + 2x - 5)$ → $8x^3 + 16x^2 - 4x - 15$

Let me know if you need further details or have any questions.

5 Related Questions:

1. How can you use the distributive property to expand binomials?
2. What are the common patterns when expanding polynomials?
3. How do you factor quadratic polynomials?
4. What is the significance of like terms in polynomial operations?
5. Can synthetic division be used to simplify complex polynomial division?

Tip: Always combine like terms after expanding polynomials to simplify the final expression efficiently.