Factorise:
` ax ^(2) + by^(2) + bx^(2) + ay^(2)`

Answer

To factorise the expression \( ax^2 + by^2 + bx^2 + ay^2 \), we can follow these steps:

### Step 1: Rearrange the terms
We can rearrange the terms in the expression for easier grouping:
\[ ax^2 + bx^2 + ay^2 + by^2 \]

### Step 2: Group the terms
Now, we can group the first two terms together and the last two terms together:
\[ (ax^2 + bx^2) + (ay^2 + by^2) \]

### Step 3: Factor out the common factors
In the first group \( ax^2 + bx^2 \), we can factor out \( x^2 \):
\[ x^2(a + b) \]
In the second group \( ay^2 + by^2 \), we can factor out \( y^2 \):
\[ y^2(a + b) \]

### Step 4: Combine the factored terms
Now we can combine the factored terms:
\[ x^2(a + b) + y^2(a + b) \]

### Step 5: Factor out the common binomial
Since \( (a + b) \) is common in both terms, we can factor it out:
\[ (a + b)(x^2 + y^2) \]

### Final Answer
Thus, the factorised form of the expression \( ax^2 + by^2 + bx^2 + ay^2 \) is:
\[ (a + b)(x^2 + y^2) \]

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