📚 Expanding Brackets: Methods of Removing Brackets in Polynomial Multiplication | 整式乘法：去括号展开的方法
Expanding brackets, also known as removing parentheses, is the process of rewriting a product of algebraic expressions as a sum of individual terms. It is a core skill in A-Level Mathematics, appearing in differentiation, integration, curve sketching, and equation solving.
去括号展开，是指将若干个代数式相乘的算式改写为各项之和的化简过程。这是 A-Level 数学的核心技能，在微分、积分、函数图像以及解方程中都会反复用到。
1. The Distributive Law | 分配律
The distributive law states that a(b + c) = ab + ac. Every term inside the bracket must be multiplied by the term outside it.
分配律表述为 a(b + c) = ab + ac。括号外的每一项都必须与括号内的每一项分别相乘。
a(b + c) = ab + ac
For example, 4(2x² + 3x − 1) = 8x² + 12x − 4. Notice that the constant term −1 is also multiplied by 4.
例如，4(2x² + 3x − 1) = 8x² + 12x − 4。注意常数项 −1 也要乘以 4。
When the coefficient outside the bracket is negative, take particular care with signs: −3(2x − 5) = −6x + 15.
当括号前系数为负时，需特别注意符号变化：−3(2x − 5) = −6x + 15。
The key rule is that the multiplication must apply to every term inside, never just the first term encountered.
关键法则在于：相乘操作必须作用于括号内的每一个项，切不可只乘以第一个遇到的项。
2. The FOIL Method for Two Binomials | 二项式乘法中的 FOIL 方法
When multiplying two binomials, the FOIL method guarantees that all four products are found. FOIL stands for First, Outer, Inner, and Last.
当两个二项式相乘时，FOIL 方法可以保证四个乘积全部被找到。FOIL 分别代表 First（首项）、Outer（外项）、Inner（内项）与 Last（末项）。
To expand (x + 3)(x + 5), follow these steps:
展开 (x + 3)(x + 5) 时，按以下步骤操作：
- First: x × x = x²
- Outer: x × 5 = 5x
- Inner: 3 × x = 3x
- Last: 3 × 5 = 15
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
After applying FOIL, always collect like terms. Here 5x and 3x combine to give 8x.
使用 FOIL 之后，务必合并同类项。本例中 5x 与 3x 合并为 8x。
3. Perfect Square Formulas | 完全平方公式
Two important expansions appear so often that they should be memorised entirely.
两个非常重要的展开式出现频率极高，应当完整记忆。
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
For instance, (3x + 2)² = (3x)² + 2(3x)(2) + 2² = 9x² + 12x + 4.
例如，(3x + 2)² = (3x)² + 2(3x)(2) + 2² = 9x² + 12x + 4。
A very common error is to write (a + b)² = a² + b². This is incorrect because it omits the middle term 2ab.
一个常见错误是写成 (a + b)² = a² + b²。这是错误的，因为它漏掉了中间项 2ab。
4. Difference of Two
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