Factorise:
`ab (x ^(2) + y ^(2)) - xy (a ^(2) + b ^(2))`

Answer

To factorise the expression \( ab(x^2 + y^2) - xy(a^2 + b^2) \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ ab(x^2 + y^2) - xy(a^2 + b^2) \]

### Step 2: Expand the Terms
Now, we expand both terms:
\[ = abx^2 + aby^2 - (xya^2 + xyb^2) \]
This gives us:
\[ = abx^2 + aby^2 - xya^2 - xyb^2 \]

### Step 3: Group the Terms
Next, we group the terms in pairs:
\[ = (abx^2 - xya^2) + (aby^2 - xyb^2) \]

### Step 4: Factor Out Common Factors
Now we factor out the common factors from each group:
1. From the first group \( abx^2 - xya^2 \), we can factor out \( x \):
\[ = x(abx - ya^2) \]
2. From the second group \( aby^2 - xyb^2 \), we can factor out \( y \):
\[ = y(aby - xb^2) \]

### Step 5: Rearranging and Factoring
Now we can rearrange and factor out the common binomial factor:
\[ = x(a^2b - yb^2) + y(ab - xb) \]

### Step 6: Final Factorization
Now we can see that both terms have a common factor of \( (ax - by) \):
\[ = (ax - by)(b + a) \]

Thus, the final factorised form of the expression \( ab(x^2 + y^2) - xy(a^2 + b^2) \) is:
\[ (ax - by)(ab + xy) \]