Factorise :
`a^(2)(b+c) - (b+c)^(3)`

Answer

To factorise the expression \( a^2(b+c) - (b+c)^3 \), we can follow these steps:

### Step 1: Identify the common factor
Notice that both terms in the expression share a common factor of \( (b+c) \). We can factor this out from the expression.

\[ a^2(b+c) - (b+c)^3 = (b+c)(a^2 - (b+c)^2) \]

### Step 2: Recognize the difference of squares
The expression \( a^2 - (b+c)^2 \) is a difference of squares. We can use the identity \( x^2 - y^2 = (x+y)(x-y) \) where \( x = a \) and \( y = (b+c) \).

\[ a^2 - (b+c)^2 = (a + (b+c))(a - (b+c)) \]

### Step 3: Substitute back into the expression
Now we can substitute this back into our factored expression from Step 1:

\[ (b+c)(a + (b+c))(a - (b+c)) \]

### Step 4: Simplify the expression
We can rewrite the expression to make it clearer:

\[ (b+c)(a + b + c)(a - b - c) \]

### Final Answer
Thus, the fully factored form of the expression \( a^2(b+c) - (b+c)^3 \) is:

\[ (b+c)(a + b + c)(a - b - c) \]

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