Students can use this sheet to practice expanding binomials like (x + 9)(x - 8).

Problem Overview:

The task is to find the product of each pair of factors given in the form \((x + a)(x + b)\) or \((x - a)(x - b)\). This involves expanding each expression using the distributive property (also known as the FOIL method for binomials).

General Formula:

For any two binomials \((x + a)(x + b)\):
\[
(x + a)(x + b) = x^2 + (a + b)x + ab
\]

For \((x - a)(x - b)\):
\[
(x - a)(x - b) = x^2 - (a + b)x + ab
\]

For \((x + a)(x - b)\):
\[
(x + a)(x - b) = x^2 + (a - b)x - ab
\]

Solution:

We will solve each problem step by step.

---

#### 1. \((x + 8)(x - 2)\)
Using the formula \((x + a)(x - b) = x^2 + (a - b)x - ab\):
- \(a = 8\), \(b = 2\)
- \(a - b = 8 - 2 = 6\)
- \(-ab = -(8 \cdot 2) = -16\)

So,
\[
(x + 8)(x - 2) = x^2 + 6x - 16
\]

#### 2. \((x + 4)(x - 9)\)
Using the formula \((x + a)(x - b) = x^2 + (a - b)x - ab\):
- \(a = 4\), \(b = 9\)
- \(a - b = 4 - 9 = -5\)
- \(-ab = -(4 \cdot 9) = -36\)

So,
\[
(x + 4)(x - 9) = x^2 - 5x - 36
\]

#### 3. \((x - 7)(x + 4)\)
Using the formula \((x - a)(x + b) = x^2 + (b - a)x - ab\):
- \(a = 7\), \(b = 4\)
- \(b - a = 4 - 7 = -3\)
- \(-ab = -(7 \cdot 4) = -28\)

So,
\[
(x - 7)(x + 4) = x^2 - 3x - 28
\]

#### 4. \((x - 9)(x + 4)\)
Using the formula \((x - a)(x + b) = x^2 + (b - a)x - ab\):
- \(a = 9\), \(b = 4\)
- \(b - a = 4 - 9 = -5\)
- \(-ab = -(9 \cdot 4) = -36\)

So,
\[
(x - 9)(x + 4) = x^2 - 5x - 36
\]

#### 5. \((x + 9)(x + 3)\)
Using the formula \((x + a)(x + b) = x^2 + (a + b)x + ab\):
- \(a = 9\), \(b = 3\)
- \(a + b = 9 + 3 = 12\)
- \(ab = 9 \cdot 3 = 27\)

So,
\[
(x + 9)(x + 3) = x^2 + 12x + 27
\]

#### 6. \((x - 5)(x - 1)\)
Using the formula \((x - a)(x - b) = x^2 - (a + b)x + ab\):
- \(a = 5\), \(b = 1\)
- \(a + b = 5 + 1 = 6\)
- \(ab = 5 \cdot 1 = 5\)

So,
\[
(x - 5)(x - 1) = x^2 - 6x + 5
\]

#### 7. \((x - 3)(x - 3)\)
Using the formula \((x - a)(x - b) = x^2 - (a + b)x + ab\):
- \(a = 3\), \(b = 3\)
- \(a + b = 3 + 3 = 6\)
- \(ab = 3 \cdot 3 = 9\)

So,
\[
(x - 3)(x - 3) = x^2 - 6x + 9
\]

#### 8. \((x - 9)(x + 2)\)
Using the formula \((x - a)(x + b) = x^2 + (b - a)x - ab\):
- \(a = 9\), \(b = 2\)
- \(b - a = 2 - 9 = -7\)
- \(-ab = -(9 \cdot 2) = -18\)

So,
\[
(x - 9)(x + 2) = x^2 - 7x - 18
\]

#### 9. \((x - 1)(x + 6)\)
Using the formula \((x - a)(x + b) = x^2 + (b - a)x - ab\):
- \(a = 1\), \(b = 6\)
- \(b - a = 6 - 1 = 5\)
- \(-ab = -(1 \cdot 6) = -6\)

So,
\[
(x - 1)(x + 6) = x^2 + 5x - 6
\]

#### 10. \((x + 7)(x - 4)\)
Using the formula \((x + a)(x - b) = x^2 + (a - b)x - ab\):
- \(a = 7\), \(b = 4\)
- \(a - b = 7 - 4 = 3\)
- \(-ab = -(7 \cdot 4) = -28\)

So,
\[
(x + 7)(x - 4) = x^2 + 3x - 28
\]

#### 11. \((x + 9)(x - 8)\)
Using the formula \((x + a)(x - b) = x^2 + (a - b)x - ab\):
- \(a = 9\), \(b = 8\)
- \(a - b = 9 - 8 = 1\)
- \(-ab = -(9 \cdot 8) = -72\)

So,
\[
(x + 9)(x - 8) = x^2 + x - 72
\]

#### 12. \((x + 6)(x - 3)\)
Using the formula \((x + a)(x - b) = x^2 + (a - b)x - ab\):
- \(a = 6\), \(b = 3\)
- \(a - b = 6 - 3 = 3\)
- \(-ab = -(6 \cdot 3) = -18\)

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

#### 13. \((x + 5)(x - 9)\)
Using the formula \((x + a)(x - b) = x^2 + (a - b)x - ab\):
- \(a = 5\), \(b = 9\)
- \(a - b = 5 - 9 = -4\)
- \(-ab = -(5 \cdot 9) = -45\)

So,
\[
(x + 5)(x - 9) = x^2 - 4x - 45
\]

#### 14. \((x + 1)(x - 5)\)
Using the formula \((x + a)(x - b) = x^2 + (a - b)x - ab\):
- \(a = 1\), \(b = 5\)
- \(a - b = 1 - 5 = -4\)
- \(-ab = -(1 \cdot 5) = -5\)

So,
\[
(x + 1)(x - 5) = x^2 - 4x - 5
\]

#### 15. \((x + 7)(x + 4)\)
Using the formula \((x + a)(x + b) = x^2 + (a + b)x + ab\):
- \(a = 7\), \(b = 4\)
- \(a + b = 7 + 4 = 11\)
- \(ab = 7 \cdot 4 = 28\)

So,
\[
(x + 7)(x + 4) = x^2 + 11x + 28
\]

#### 16. \((x - 2)(x - 2)\)
Using the formula \((x - a)(x - b) = x^2 - (a + b)x + ab\):
- \(a = 2\), \(b = 2\)
- \(a + b = 2 + 2 = 4\)
- \(ab = 2 \cdot 2 = 4\)

So,
\[
(x - 2)(x - 2) = x^2 - 4x + 4
\]

#### 17. \((x - 9)(x + 2)\)
Using the formula \((x - a)(x + b) = x^2 + (b - a)x - ab\):
- \(a = 9\), \(b = 2\)
- \(b - a = 2 - 9 = -7\)
- \(-ab = -(9 \cdot 2) = -18\)

So,
\[
(x - 9)(x + 2) = x^2 - 7x - 18
\]

#### 18. \((x + 6)(x + 7)\)
Using the formula \((x + a)(x + b) = x^2 + (a + b)x + ab\):
- \(a = 6\), \(b = 7\)
- \(a + b = 6 + 7 = 13\)
- \(ab = 6 \cdot 7 = 42\)

So,
\[
(x + 6)(x + 7) = x^2 + 13x + 42
\]

#### 19. \((x + 2)(x + 1)\)
Using the formula \((x + a)(x + b) = x^2 + (a + b)x + ab\):
- \(a = 2\), \(b = 1\)
- \(a + b = 2 + 1 = 3\)
- \(ab = 2 \cdot 1 = 2\)

So,
\[
(x + 2)(x + 1) = x^2 + 3x + 2
\]

#### 20. \((x - 4)(x + 4)\)
Using the formula \((x - a)(x + b) = x^2 + (b - a)x - ab\):
- \(a = 4\), \(b = 4\)
- \(b - a = 4 - 4 = 0\)
- \(-ab = -(4 \cdot 4) = -16\)

So,
\[
(x - 4)(x + 4) = x^2 - 16
\]

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad x^2 + 6x - 16 \\
2. & \quad x^2 - 5x - 36 \\
3. & \quad x^2 - 3x - 28 \\
4. & \quad x^2 - 5x - 36 \\
5. & \quad x^2 + 12x + 27 \\
6. & \quad x^2 - 6x + 5 \\
7. & \quad x^2 - 6x + 9 \\
8. & \quad x^2 - 7x - 18 \\
9. & \quad x^2 + 5x - 6 \\
10. & \quad x^2 + 3x - 28 \\
11. & \quad x^2 + x - 72 \\
12. & \quad x^2 + 3x - 18 \\
13. & \quad x^2 - 4x - 45 \\
14. & \quad x^2 - 4x - 5 \\
15. & \quad x^2 + 11x + 28 \\
16. & \quad x^2 - 4x + 4 \\
17. & \quad x^2 - 7x - 18 \\
18. & \quad x^2 + 13x + 42 \\
19. & \quad x^2 + 3x + 2 \\
20. & \quad x^2 - 16 \\
\end{aligned}
}
\]