Fractorise:
`(ax + by) ^(2) + (bx - ay)^(2)`

Answer

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

### Step 1: Expand the squares
We start by expanding both squares using the formula \((a + b)^2 = a^2 + b^2 + 2ab\) and \((a - b)^2 = a^2 + b^2 - 2ab\).

\[ (ax + by)^2 = a^2x^2 + b^2y^2 + 2abxy \]
\[ (bx - ay)^2 = b^2x^2 + a^2y^2 - 2abxy \]

### Step 2: Combine the expanded expressions
Now, we combine the two expanded expressions:

\[ (ax + by)^2 + (bx - ay)^2 = (a^2x^2 + b^2y^2 + 2abxy) + (b^2x^2 + a^2y^2 - 2abxy) \]

### Step 3: Simplify the expression
Next, we simplify the combined expression:

\[ = a^2x^2 + b^2y^2 + 2abxy + b^2x^2 + a^2y^2 - 2abxy \]

The \(2abxy\) and \(-2abxy\) cancel each other out:

\[ = a^2x^2 + b^2y^2 + b^2x^2 + a^2y^2 \]

### Step 4: Rearrange the terms
We can rearrange the terms:

\[ = a^2x^2 + b^2x^2 + a^2y^2 + b^2y^2 \]

### Step 5: Factor out common terms
Now, we can factor out the common terms:

\[ = (a^2 + b^2)(x^2 + y^2) \]

### Final Answer
Thus, the factorized form of the expression \((ax + by)^2 + (bx - ay)^2\) is:

\[ (a^2 + b^2)(x^2 + y^2) \]

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