Factorising double brackets is a crucial algebraic skill that unlocks the door to solving complex equations and simplifying expressions. In this comprehensive guide, we will delve into the art of factorising double brackets, providing you with a step-by-step approach to tackle even the most challenging problems.

**Understanding the Basics**

Before we dive into the process, let’s refresh our understanding of factors. A factor is a number or expression that divides another number or expression without leaving a remainder. In the context of factorising double brackets, we are essentially reversing the process of multiplying two expressions to obtain a single expression.

**Step 1: Identifying the Common Factors**

Begin by identifying any common factors shared by the terms within the double brackets. Look for common variables or numbers that can be factored out. This step sets the foundation for further simplification.

**Step 2: Using the Distributive Property**

The distributive property is the key to factorising double brackets. Multiply the common factor (found in Step 1) with each term inside the brackets. This process distributes the common factor across both terms.

**Step 3: Checking for Further Factorisation**

After applying the distributive property, examine the result to check if there are any opportunities for further factorisation. Some expressions might still have common factors that can be factored out, leading to a more simplified form.

**Step 4: The Difference of Two Squares**

In some cases, you might encounter expressions that resemble the difference of two squares, such as (a^2 – b^2). Remember the formula: (a^2 – b^2) = (a + b)(a – b). Use this rule to factorise such expressions swiftly.

**Step 5: Completing the Process**

Continue applying the steps until you can no longer simplify the expression further. By the end of this process, you will have successfully factorised the double brackets into their simplest form.

**Example:**

Let’s illustrate the process with an example:

Factorise: (3x + 6y)

**Step 1:** The common factor here is 3.

**Step 2:** Using the distributive property: 3(x + 2y)

**Step 3:** No further factorisation possible.

**Conclusion**

Congratulations! You have now mastered the art of factorising double brackets. This powerful algebraic technique will prove invaluable in solving equations, simplifying expressions, and tackling advanced mathematical problems. With practice, you will gain confidence in handling even the most intricate factorisation challenges.

Whether you’re a student aiming to excel in algebra or an educator guiding young minds, understanding factorising double brackets opens the gateway to mathematical success. So, embrace this skill, practice diligently, and unlock a world of mathematical possibilities!