 
B. SIMPLIFY THE POLYNOMIALS
 
8.    Multiply:      (7x2 – 8) (4x2 – 6)  ________________________ 
  
9.    Multiply:      (3a + 5) (3a – 5)  _________________________
 
10. (x + 6) (x – 3) (x + 1) 11.     15x^2+ 19x – 4   ____________________________

             3x +  8 
 
12.    Divide the Polynomials: 2a^3+ 8a^2+ 5a – 6a  +  2
 

Let's simplify each of the polynomials one by one:

### 8. Multiply: \( (7x^2 - 8)(4x^2 - 6) \)

To multiply these polynomials, we apply the distributive property (also known as the FOIL method for binomials):

\[
(7x^2 - 8)(4x^2 - 6) = 7x^2 \cdot 4x^2 + 7x^2 \cdot (-6) - 8 \cdot 4x^2 - 8 \cdot (-6)
\]

Calculating each term:

1. \(7x^2 \cdot 4x^2 = 28x^4\)
2. \(7x^2 \cdot (-6) = -42x^2\)
3. \((-8) \cdot 4x^2 = -32x^2\)
4. \((-8) \cdot (-6) = 48\)

Combine all the terms:

\[
28x^4 - 42x^2 - 32x^2 + 48 = 28x^4 - 74x^2 + 48
\]

**Simplified Result:** \(28x^4 - 74x^2 + 48\)

### 9. Multiply: \( (3a + 5)(3a - 5) \)

This is a difference of squares formula: \( (a + b)(a - b) = a^2 - b^2 \).

\[
(3a + 5)(3a - 5) = (3a)^2 - 5^2 = 9a^2 - 25
\]

**Simplified Result:** \(9a^2 - 25\)

### 10. Multiply: \( (x + 6)(x - 3)(x + 1) \)

Let's multiply the first two binomials first:

\[
(x + 6)(x - 3) = x^2 - 3x + 6x - 18 = x^2 + 3x - 18
\]

Now, multiply the result by the third binomial:

\[
(x^2 + 3x - 18)(x + 1) = x^3 + x^2 + 3x^2 + 3x - 18x - 18
\]

Combine like terms:

\[
x^3 + 4x^2 - 15x - 18
\]

**Simplified Result:** \(x^3 + 4x^2 - 15x - 18\)

### 11. Divide: \( \frac{15x^2 + 19x - 4}{3x + 8} \)

To divide polynomials, we use polynomial long division or synthetic division. Here, we'll use polynomial long division:

1. Divide the first term of the dividend (\(15x^2\)) by the first term of the divisor (\(3x\)): \( \frac{15x^2}{3x} = 5x \).
   
2. Multiply \(5x\) by \(3x + 8\): \(5x(3x + 8) = 15x^2 + 40x\).

3. Subtract: 

\[
(15x^2 + 19x - 4) - (15x^2 + 40x) = -21x - 4
\]

4. Divide the next term: \(\frac{-21x}{3x} = -7\).

5. Multiply: \(-7(3x + 8) = -21x - 56\).

6. Subtract:

\[
(-21x - 4) - (-21x - 56) = 52
\]

So, the quotient is \(5x - 7\) and the remainder is \(52\). 

**Simplified Result:** \(5x - 7 + \frac{52}{3x + 8}\)

### 12. Divide the Polynomials: \( \frac{2a^3 + 8a^2 + 5a - 6}{a + 2} \)

Let's use polynomial long division for this division:

1. Divide the first term: \(\frac{2a^3}{a} = 2a^2\).

2. Multiply: \(2a^2(a + 2) = 2a^3 + 4a^2\).

3. Subtract:

\[
(2a^3 + 8a^2 + 5a - 6) - (2a^3 + 4a^2) = 4a^2 + 5a - 6
\]

4. Divide the next term: \(\frac{4a^2}{a} = 4a\).

5. Multiply: \(4a(a + 2) = 4a^2 + 8a\).

6. Subtract:

\[
(4a^2 + 5a - 6) - (4a^2 + 8a) = -3a - 6
\]

7. Divide the next term: \(\frac{-3a}{a} = -3\).

8. Multiply: \(-3(a + 2) = -3a - 6\).

9. Subtract:

\[
(-3a - 6) - (-3a - 6) = 0
\]

So, the quotient is \(2a^2 + 4a - 3\).

**Simplified Result:** \(2a^2 + 4a - 3\)

### Summary of Simplified Results:
1. \(28x^4 - 74x^2 + 48\)
2. \(9a^2 - 25\)
3. \(x^3 + 4x^2 - 15x - 18\)
4. \(5x - 7 + \frac{52}{3x + 8}\)
5. \(2a^2 + 4a - 3\)

Would you like more details or explanations on any of these steps?

### Related Questions:
1. How do you perform polynomial long division for more complicated polynomials?
2. What is the difference between polynomial long division and synthetic division?
3. Can you explain the FOIL method in detail for multiplying binomials?
4. What are special products, and how do they simplify polynomial multiplication?
5. How do you factor a polynomial after expanding it?

### Tip:
When multiplying polynomials, look for common patterns like the difference of squares or perfect square trinomials to simplify the multiplication process.

B. SIMPLIFY THE POLYNOMIALS

8. Multiply: (7x^2 – 8) (4x^2 – 6)

9. Multiply: (3a + 5) (3a – 5)

10. (x + 6) (x – 3) (x + 1)

11. 15x^2 + 19x – 4 divided by 3x + 8

12. Divide the Polynomials: 2a^3 + 8a^2 + 5a – 6 divided by a + 2

Learn how to multiply and divide polynomials like (7x^2 – 8)(4x^2 – 6) and (3a + 5)(3a – 5) using the FOIL method, difference of squares, and polynomial long division. Detailed solutions included for Grades 9-12.

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