Factorise :
`4a^(2) - (4b^(2) + 4bc + c^(2))`

Answer

To factorise the expression \( 4a^2 - (4b^2 + 4bc + c^2) \), we can follow these steps:

### Step 1: Rewrite the expression
Start by rewriting the expression to make it clearer:
\[ 4a^2 - (4b^2 + 4bc + c^2) \]

### Step 2: Distribute the negative sign
Distributing the negative sign inside the bracket gives:
\[ 4a^2 - 4b^2 - 4bc - c^2 \]

### Step 3: Identify perfect squares
Notice that \( 4a^2 \) can be written as \( (2a)^2 \) and \( 4b^2 + 4bc + c^2 \) can be recognized as a perfect square:
\[ 4b^2 + 4bc + c^2 = (2b + c)^2 \]
Thus, we can rewrite the expression as:
\[ (2a)^2 - (2b + c)^2 \]

### Step 4: Apply the difference of squares formula
Now, we can apply the difference of squares formula, which states that \( x^2 - y^2 = (x - y)(x + y) \). Here, \( x = 2a \) and \( y = 2b + c \):
\[ (2a - (2b + c))(2a + (2b + c)) \]

### Step 5: Simplify the expression
Now simplify the expressions in the brackets:
\[ (2a - 2b - c)(2a + 2b + c) \]

### Final Result
Thus, the factorised form of the expression \( 4a^2 - (4b^2 + 4bc + c^2) \) is:
\[ (2a - 2b - c)(2a + 2b + c) \]

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