Find the product 1. (2j + 3k - 4) ^ 2

2. (5a + b)(5a - b)

3. (2x * y ^ 2 + 3z)(2x * y ^ 2 - 3z)

4. (5w - 3) ^ 3

5. (3xy - 6) ^ 2

6. (2y - 8)(4y ^ 2 - 16y + 64)

7. (4xy - 3z)(16x ^ 2 * y ^ 2 + 12xyz + 9z ^ 2)

8. (5x ^ 2 * y ^ 2 + 7)(5x ^ 2 * y ^ 2 - 7)

9. (3a - 5b - 1) ^ 2

10. (7x + 3y) ^ 2

Let's solve the products one by one.

### 1. \((2j + 3k - 4)^2\)

This is a binomial squared, so we'll expand it using the formula \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\):
\[
(2j + 3k - 4)^2 = (2j)^2 + (3k)^2 + (-4)^2 + 2(2j)(3k) + 2(2j)(-4) + 2(3k)(-4)
\]
\[
= 4j^2 + 9k^2 + 16 + 12jk - 16j - 24k
\]
So, the expanded form is:
\[
4j^2 + 9k^2 + 16 + 12jk - 16j - 24k
\]

### 2. \((5a + b)(5a - b)\)

This is a difference of squares: \((x + y)(x - y) = x^2 - y^2\):
\[
(5a + b)(5a - b) = (5a)^2 - b^2
\]
\[
= 25a^2 - b^2
\]
So, the result is:
\[
25a^2 - b^2
\]

### 3. \((2x \cdot y^2 + 3z)(2x \cdot y^2 - 3z)\)

This is again a difference of squares:
\[
(2x \cdot y^2 + 3z)(2x \cdot y^2 - 3z) = (2x \cdot y^2)^2 - (3z)^2
\]
\[
= 4x^2 y^4 - 9z^2
\]
So, the result is:
\[
4x^2 y^4 - 9z^2
\]

### 4. \((5w - 3)^3\)

This is the cube of a binomial. Use the expansion formula \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\):
\[
(5w - 3)^3 = (5w)^3 - 3(5w)^2(3) + 3(5w)(3)^2 - (3)^3
\]
\[
= 125w^3 - 3(25w^2)(3) + 3(5w)(9) - 27
\]
\[
= 125w^3 - 225w^2 + 135w - 27
\]
So, the result is:
\[
125w^3 - 225w^2 + 135w - 27
\]

### 5. \((3xy - 6)^2\)

This is a binomial square, so we'll expand it using the formula \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(3xy - 6)^2 = (3xy)^2 - 2(3xy)(6) + 6^2
\]
\[
= 9x^2 y^2 - 36xy + 36
\]
So, the expanded form is:
\[
9x^2 y^2 - 36xy + 36
\]

### 6. \((2y - 8)(4y^2 - 16y + 64)\)

This requires distributing \((2y - 8)\) across \((4y^2 - 16y + 64)\):
\[
= 2y(4y^2 - 16y + 64) - 8(4y^2 - 16y + 64)
\]
\[
= 2y(4y^2) + 2y(-16y) + 2y(64) - 8(4y^2) - 8(-16y) - 8(64)
\]
\[
= 8y^3 - 32y^2 + 128y - 32y^2 + 128y - 512
\]
\[
= 8y^3 - 64y^2 + 256y - 512
\]
So, the result is:
\[
8y^3 - 64y^2 + 256y - 512
\]

### 7. \((4xy - 3z)(16x^2 y^2 + 12xyz + 9z^2)\)

We distribute \((4xy - 3z)\) across \((16x^2 y^2 + 12xyz + 9z^2)\):
\[
= 4xy(16x^2 y^2 + 12xyz + 9z^2) - 3z(16x^2 y^2 + 12xyz + 9z^2)
\]
\[
= 4xy(16x^2 y^2) + 4xy(12xyz) + 4xy(9z^2) - 3z(16x^2 y^2) - 3z(12xyz) - 3z(9z^2)
\]
\[
= 64x^3 y^3 + 48x^2 y^2 z + 36xyz^2 - 48x^2 y^2 z - 36xyz^2 - 27z^3
\]
\[
= 64x^3 y^3 - 27z^3
\]
So, the result is:
\[
64x^3 y^3 - 27z^3
\]

### 8. \((5x^2 y^2 + 7)(5x^2 y^2 - 7)\)

This is a difference of squares:
\[
(5x^2 y^2 + 7)(5x^2 y^2 - 7) = (5x^2 y^2)^2 - 7^2
\]
\[
= 25x^4 y^4 - 49
\]
So, the result is:
\[
25x^4 y^4 - 49
\]

### 9. \((3a - 5b - 1)^2\)

We'll expand it using \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\):
\[
(3a - 5b - 1)^2 = (3a)^2 + (-5b)^2 + (-1)^2 + 2(3a)(-5b) + 2(3a)(-1) + 2(-5b)(-1)
\]
\[
= 9a^2 + 25b^2 + 1 - 30ab - 6a + 10b
\]
So, the result is:
\[
9a^2 + 25b^2 + 1 - 30ab - 6a + 10b
\]

### 10. \((7x + 3y)^2\)

This is a binomial square:
\[
(7x + 3y)^2 = (7x)^2 + 2(7x)(3y) + (3y)^2
\]
\[
= 49x^2 + 42xy + 9y^2
\]
So, the result is:
\[
49x^2 + 42xy + 9y^2
\]

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Let me know if you'd like more details or have any questions!

Here are five related questions:
1. How do you expand a cube of a binomial?
2. Can you solve the product \((a - b)(a + b)\) with variables?
3. What is the expansion of \((x + y - z)^2\)?
4. How do you simplify products involving higher powers, like \((x^2 + 1)(x^2 - 1)\)?
5. What is the general formula for the difference of squares?

**Tip**: Always look for patterns like the difference of squares or cubes, as they simplify products greatly.

Find the product 1. (2j + 3k - 4) ^ 2 2. (5a + b)(5a - b) 3. (2x * y ^ 2 + 3z)(2x * y ^ 2 - 3z) 4. (5w - 3) ^ 3 5. (3xy - 6) ^ 2 6. (2y - 8)(4y ^ 2 - 16y + 64) 7. (4xy - 3z)(16x ^ 2 * y ^ 2 + 12xyz + 9z ^ 2) 8. (5x ^ 2 * y ^ 2 + 7)(5x ^ 2 * y ^ 2 - 7) 9. (3a - 5b - 1) ^ 2 10. (7x + 3y) ^ 2

This article provides detailed solutions for expanding and simplifying various polynomial expressions, including binomials, cubic expansions, and difference of squares. Topics covered include formulas like (a + b + c)^2 and (a - b)^3, suitable for students in Grades 9-12.

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