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

Answer

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

### Step 1: Rearrange the expression
We start with the expression:
\[ a^2 - b^2 - c^2 + 2bc \]
We can rearrange it as:
\[ a^2 - (b^2 - 2bc + c^2) \]

### Step 2: Recognize a perfect square
Notice that \( b^2 - 2bc + c^2 \) can be factored as a perfect square:
\[ b^2 - 2bc + c^2 = (b - c)^2 \]
So, we can rewrite our expression as:
\[ a^2 - (b - c)^2 \]

### Step 3: 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, let \( x = a \) and \( y = (b - c) \):
\[ a^2 - (b - c)^2 = (a + (b - c))(a - (b - c)) \]

### Step 4: Simplify the factors
Now we simplify the factors:
\[ = (a + b - c)(a - b + c) \]

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

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