This comprehensive guide is designed to enhance the understanding and skills of CIE IGCSE students in expanding algebraic expressions, focusing on the application of the distribution law to expand expressions involving products. Each example and practice problem is meticulously worked through with detailed equations to ensure mathematical accuracy and clarity.

**Understanding the Distribution Law**

The distribution law, or distributive property, is fundamental in algebra for expanding expressions. It is represented as $a(b + c) = ab + ac$, allowing for the multiplication of a sum by distributing the multiplier to each term within the parentheses.

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**Worked Examples**

**Example 1: Expanding a Simple Product**

To expand $3x(2x - 4y)$:

$3x \times 2x = 6x^2$

$3x \times (-4y) = -12xy$

Resulting in:

$6x^2 - 12xy$**Example 2: Expanding a More Complex Expression**

For $5y(3y^2 + 2y - 7)$:

$5y \times 3y^2 = 15y^3$

$5y \times 2y = 10y^2$

$5y \times (-7) = -35y$

Combining gives:

$15y^3 + 10y^2 - 35y$**Example 3: Expanding Double Binomials**

Expanding $(x + 3)(x - 2)$ involves:

$x^2 - 2x + 3x - 6$Simplified to:

$x^2 + x - 6$**Example 4: Expanding with Coefficients**

For $2(a + b)(2a - 3b)$, the detailed expansion is:

$4a^2 - 2ab - 6b^2$**Example 5: Expanding Triple Terms**

The expression $(2x - 3)(x^2 + x + 1)$ expands to:

$2x^3 - x^2 - x - 3$**Expanding Expressions with Multiple Terms**

Expanding algebraic expressions with multiple terms can become quite intricate. The examples below illustrate how to approach the expansion of expressions involving trinomials and binomials, showcasing the versatility of the distribution law in algebra.

**Example 6: Expanding a Trinomial by a Binomial**

Consider expanding the expression $(x^2 + 2x + 1)(x - 3)$:

This involves distributing each term in the trinomial across the binomial:

$x^3 - 3x^2 + 2x^2 - 6x + x - 3$Simplifying, we get:

$x^3 - x^2 - 5x - 3$**Example 7: Expanding Two Trinomials**

Expanding the expression $(2x^2 - x + 3)(x^2 + x - 1)$ involves a more complex process, given both multiplicands are trinomials:

The detailed expansion results in:

$2x^4 + x^3 - 2x^2 + 3x^2 + x^2 + x - 2x + 3x - 3$Upon combining like terms and simplifying, the final expression is:

$2x^4 + x^3 + 4x - 3$