Algebra worksheet focusing on simplifying, expanding, and factorizing algebraic expressions.

Problem Overview:

The task involves simplifying, expanding, and factorizing algebraic expressions. Let's solve each section step by step.

---

Section A: Simplify the expressions

#### 1) \( a \times b \)
- Simplification: \( ab \)

#### 2) \( 3s \times 8t \)
- Simplification: \( 3 \times 8 \times s \times t = 24st \)

#### 3) \( x \times x \)
- Simplification: \( x^2 \)

#### 4) \( 2y \times 4y \)
- Simplification: \( 2 \times 4 \times y \times y = 8y^2 \)

#### 5) \( 9pq \times p \)
- Simplification: \( 9 \times p \times q \times p = 9p^2q \)

#### 6) \( 7cd \times 11d \)
- Simplification: \( 7 \times 11 \times c \times d \times d = 77cd^2 \)

#### 7) \( f \times f \times f \)
- Simplification: \( f^3 \)

#### 8) \( -4a \times 2 \)
- Simplification: \( -4 \times 2 \times a = -8a \)

#### 9) \( 12g \times -5g^2h \)
- Simplification: \( 12 \times -5 \times g \times g^2 \times h = -60g^3h \)

#### 10) \( -7k^2 \times -3k \)
- Simplification: \( -7 \times -3 \times k^2 \times k = 21k^3 \)

---

Section B: Expand the brackets

#### 1) \( 2(3x + 4) \)
- Expansion: \( 2 \cdot 3x + 2 \cdot 4 = 6x + 8 \)

#### 2) \( 4(6y - 5) \)
- Expansion: \( 4 \cdot 6y + 4 \cdot (-5) = 24y - 20 \)

#### 3) \( 8(6 + 2t) \)
- Expansion: \( 8 \cdot 6 + 8 \cdot 2t = 48 + 16t \)

#### 4) \( 5x(y + 4) \)
- Expansion: \( 5x \cdot y + 5x \cdot 4 = 5xy + 20x \)

#### 5) \( 6a(3 - a) \)
- Expansion: \( 6a \cdot 3 + 6a \cdot (-a) = 18a - 6a^2 \)

#### 6) \( 7g(3g - 12) \)
- Expansion: \( 7g \cdot 3g + 7g \cdot (-12) = 21g^2 - 84g \)

#### 7) \( 8u(5 - uv) \)
- Expansion: \( 8u \cdot 5 + 8u \cdot (-uv) = 40u - 8u^2v \)

#### 8) \( 3e(4ef - 6e) \)
- Expansion: \( 3e \cdot 4ef + 3e \cdot (-6e) = 12e^2f - 18e^2 \)

#### 9) \( -4(7h - k) \)
- Expansion: \( -4 \cdot 7h + (-4) \cdot (-k) = -28h + 4k \)

#### 10) \( -3d(a + 2b - 9d) \)
- Expansion: \( -3d \cdot a + (-3d) \cdot 2b + (-3d) \cdot (-9d) = -3ad - 6bd + 27d^2 \)

---

Section C: Expand and simplify the expressions

#### 1) \( 8(6 + 3p) + 5p \)
- Expansion: \( 8 \cdot 6 + 8 \cdot 3p + 5p = 48 + 24p + 5p \)
- Simplification: \( 48 + 29p \)

#### 2) \( 3(x + 3) - 11 \)
- Expansion: \( 3 \cdot x + 3 \cdot 3 - 11 = 3x + 9 - 11 \)
- Simplification: \( 3x - 2 \)

#### 3) \( 5 + 6(2s + 5) \)
- Expansion: \( 5 + 6 \cdot 2s + 6 \cdot 5 = 5 + 12s + 30 \)
- Simplification: \( 35 + 12s \)

#### 4) \( m + 7(6m - 4) \)
- Expansion: \( m + 7 \cdot 6m + 7 \cdot (-4) = m + 42m - 28 \)
- Simplification: \( 43m - 28 \)

#### 5) \( 4b + 2(2 - 5b) \)
- Expansion: \( 4b + 2 \cdot 2 + 2 \cdot (-5b) = 4b + 4 - 10b \)
- Simplification: \( -6b + 4 \)

#### 6) \( 1 - 2(9w + 11) \)
- Expansion: \( 1 - 2 \cdot 9w - 2 \cdot 11 = 1 - 18w - 22 \)
- Simplification: \( -18w - 21 \)

#### 7) \( 2n - 10(2 - 6n) + 25n \)
- Expansion: \( 2n - 10 \cdot 2 - 10 \cdot (-6n) + 25n = 2n - 20 + 60n + 25n \)
- Simplification: \( 87n - 20 \)

#### 8) \( 5(9y + 8) + 2(3 - y) \)
- Expansion: \( 5 \cdot 9y + 5 \cdot 8 + 2 \cdot 3 + 2 \cdot (-y) = 45y + 40 + 6 - 2y \)
- Simplification: \( 43y + 46 \)

#### 9) \( 4(2j + 4) - (12 - j) \)
- Expansion: \( 4 \cdot 2j + 4 \cdot 4 - 12 + j = 8j + 16 - 12 + j \)
- Simplification: \( 9j + 4 \)

#### 10) \( 8x(2 + 3x) - 2x(3 + x) \)
- Expansion: \( 8x \cdot 2 + 8x \cdot 3x - 2x \cdot 3 - 2x \cdot x = 16x + 24x^2 - 6x - 2x^2 \)
- Simplification: \( 22x^2 + 10x \)

---

Section D: Factorize the following expressions

#### 1) \( 3h - 9 \)
- Factorization: \( 3(h - 3) \)

#### 2) \( 15w + 25 \)
- Factorization: \( 5(3w + 5) \)

#### 3) \( 16 - 22n \)
- Factorization: \( 2(8 - 11n) \)

#### 4) \( ab + a \)
- Factorization: \( a(b + 1) \)

#### 5) \( 9jk + 21k \)
- Factorization: \( 3k(3j + 7) \)

#### 6) \( 12h - 16gh \)
- Factorization: \( 4h(3 - 4g) \)

#### 7) \( 2x^2 - x \)
- Factorization: \( x(2x - 1) \)

#### 8) \( 24c^2d + 32c^2 \)
- Factorization: \( 8c^2(3d + 4) \)

#### 9) \( 12q + 8p - 24 \)
- Factorization: \( 4(3q + 2p - 6) \)

#### 10) \( 6mn - 27m^2 + 9m \)
- Factorization: \( 3m(2n - 9m + 3) \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{Section A:} & 1) \, ab, \, 2) \, 24st, \, 3) \, x^2, \, 4) \, 8y^2, \, 5) \, 9p^2q, \\
& 6) \, 77cd^2, \, 7) \, f^3, \, 8) \, -8a, \, 9) \, -60g^3h, \, 10) \, 21k^3 \\
\text{Section B:} & 1) \, 6x + 8, \, 2) \, 24y - 20, \, 3) \, 48 + 16t, \, 4) \, 5xy + 20x, \, 5) \, 18a - 6a^2, \\
& 6) \, 21g^2 - 84g, \, 7) \, 40u - 8u^2v, \, 8) \, 12e^2f - 18e^2, \, 9) \, -28h + 4k, \, 10) \, -3ad - 6bd + 27d^2 \\
\text{Section C:} & 1) \, 48 + 29p, \, 2) \, 3x - 2, \, 3) \, 35 + 12s, \, 4) \, 43m - 28, \, 5) \, -6b + 4, \\
& 6) \, -18w - 21, \, 7) \, 87n - 20, \, 8) \, 43y + 46, \, 9) \, 9j + 4, \, 10) \, 22x^2 + 10x \\
\text{Section D:} & 1) \, 3(h - 3), \, 2) \, 5(3w + 5), \, 3) \, 2(8 - 11n), \, 4) \, a(b + 1), \, 5) \, 3k(3j + 7), \\
& 6) \, 4h(3 - 4g), \, 7) \, x(2x - 1), \, 8) \, 8c^2(3d + 4), \, 9) \, 4(3q + 2p - 6), \, 10) \, 3m(2n - 9m + 3)
\end{array}
}
\]