Question

find each product.

1. (6x + 2)(7x - 7)
2. (3m + 6)(3m + 3)
3. (4b - 7)(4b - 2)
4. (3x + 4)(3x - 4)
5. (2v - 1)(8v - 1)
6. (n + 5)(8n + 7)

Explanation:

Step1: Solve (6x + 2)(7x - 7)

Use the distributive property (FOIL method):
First: \(6x \times 7x = 42x^2\)
Outer: \(6x \times (-7) = -42x\)
Inner: \(2 \times 7x = 14x\)
Last: \(2 \times (-7) = -14\)
Combine like terms: \(42x^2 - 42x + 14x - 14 = 42x^2 - 28x - 14\)

Step2: Solve (3n + 6)(3n + 3)

FOIL method:
First: \(3n \times 3n = 9n^2\)
Outer: \(3n \times 3 = 9n\)
Inner: \(6 \times 3n = 18n\)
Last: \(6 \times 3 = 18\)
Combine like terms: \(9n^2 + 9n + 18n + 18 = 9n^2 + 27n + 18\)

Step3: Solve (4b - 7)(4b - 2)

FOIL method:
First: \(4b \times 4b = 16b^2\)
Outer: \(4b \times (-2) = -8b\)
Inner: \(-7 \times 4b = -28b\)
Last: \(-7 \times (-2) = 14\)
Combine like terms: \(16b^2 - 8b - 28b + 14 = 16b^2 - 36b + 14\)

Step4: Solve (3x + 4)(3x - 4)

This is a difference of squares: \((a + b)(a - b) = a^2 - b^2\), where \(a = 3x\), \(b = 4\)
So, \((3x)^2 - 4^2 = 9x^2 - 16\)

Step5: Solve (2v - 1)(8v - 1)

FOIL method:
First: \(2v \times 8v = 16v^2\)
Outer: \(2v \times (-1) = -2v\)
Inner: \(-1 \times 8v = -8v\)
Last: \(-1 \times (-1) = 1\)
Combine like terms: \(16v^2 - 2v - 8v + 1 = 16v^2 - 10v + 1\)

Step6: Solve (n + 5)(8n + 7)

FOIL method:
First: \(n \times 8n = 8n^2\)
Outer: \(n \times 7 = 7n\)
Inner: \(5 \times 8n = 40n\)
Last: \(5 \times 7 = 35\)
Combine like terms: \(8n^2 + 7n + 40n + 35 = 8n^2 + 47n + 35\)

Answer:

1. \(42x^2 - 28x - 14\)
2. \(9n^2 + 27n + 18\)
3. \(16b^2 - 36b + 14\)
4. \(9x^2 - 16\)
5. \(16v^2 - 10v + 1\)
6. \(8n^2 + 47n + 35\)