Question 1

Sure! Let's go through each question one by one and calculate the requested products.

### Question 1
To find the product \((4x^3 + 2x^2)(6x - 9)\), we will use the distributive property (also called the FOIL method for binomials).

1. \(4x^3(6x) = 24x^4\)
2. \(4x^3(-9) = -36x^3\)
3. \(2x^2(6x) = 12x^3\)
4. \(2x^2(-9) = -18x^2\)

Now, we combine these results:
\[
(4x^3 + 2x^2)(6x - 9) = 24x^4 + (-36x^3 + 12x^3) + (-18x^2) = 24x^4 - 24x^3 - 18x^2
\]

Thus,
\[
(4x^3 + 2x^2)(6x - 9) = 24x^4 - 24x^3 - 18x^2
\]

### Question 2
We need to fill in the table for \((5y^2 - 6y)(7y^7 - y^3)\).

Using distribution:
1. \(5y^2(7y^7) = 35y^9\)
2. \(5y^2(-y^3) = -5y^5\)
3. \(-6y(7y^7) = -42y^8\)
4. \(-6y(-y^3) = 6y^4\)

Now, we combine:
\[
(5y^2 - 6y)(7y^7 - y^3) = 35y^9 - 42y^8 - 5y^5 + 6y^4
\]

Thus,
\[
(5y^2 - 6y)(7y^7 - y^3) = 35y^9 - 42y^8 - 5y^5 + 6y^4
\]

### Question 3
Using the Distributive Property to multiply \(3x^2(2x^4 - 15x)\):
1. \(3x^2(2x^4) = 6x^6\)
2. \(3x^2(-15x) = -45x^3\)

So,
\[
3x^2(2x^4 - 15x) = 6x^6 - 45x^3
\]

### Question 4
Now we need to compute \((3x^2 - 8)(4x^2 + 7)\):
1. \(3x^2(4x^2) = 12x^4\)
2. \(3x^2(7) = 21x^2\)
3. \(-8(4x^2) = -32x^2\)
4. \(-8(7) = -56\)

Combining:
\[
(3x^2 - 8)(4x^2 + 7) = 12x^4 + (21x^2 - 32x^2) - 56 = 12x^4 - 11x^2 - 56
\]

### Question 5
Now we will find the product of \((-3x^5 - 4x^4)(7x^2 - 2x + 6)\):
1. \(-3x^5(7x^2) = -21x^7\)
2. \(-3x^5(-2x) = 6x^6\)
3. \(-3x^5(6) = -18x^5\)
4. \(-4x^4(7x^2) = -28x^6\)
5. \(-4x^4(-2x) = 8x^5\)
6. \(-4x^4(6) = -24x^4\)

Now we combine all the terms:
\[
-21x^7 + (6x^6 - 28x^6) + (-18x^5 + 8x^5) - 24x^4 = -21x^7 - 22x^6 - 10x^5 - 24x^4
\]

### Summary of Answers
1) \( (4x^3 + 2x^2)(6x - 9) = 24x^4 - 24x^3 - 18x^2 \)

2) \( (5y^2 - 6y)(7y^7 - y^3) = 35y^9 - 42y^8 - 5y^5 + 6y^4 \)

3) \( 3x^2(2x^4 - 15x) = 6x^6 - 45x^3 \)

4) \( (3x^2 - 8)(4x^2 + 7) = 12x^4 - 11x^2 - 56 \)

5) \( (-3x^5 - 4x^4)(7x^2 - 2x + 6) = -21x^7 - 22x^6 - 10x^5 - 24x^4 \)