Question

Factorise the following:

(1 – x²) (1 – y²) + 4xy

Answer 1

Given: (1 – x²) (1 – y²) + 4xy

Step-wise calculation:

Step 1: Expand (1 – x²)(1 – y²)

(1 – x²)(1 – y²) 

= 1 – y² – x² + x²y²

So the expression becomes:

1 – x² – y² + x²y² + 4xy

Rearrange:

x²y² – x² – y² + 1 + 4xy

Step 2: Recognize a useful identity

Notice that:

x²y² – x² – y² + 1 

= (xy – 1)² – (x – y)²

Check:

(xy – 1)² = x²y² – 2xy + 1

(x – y)² = x² – 2xy + y²

Subtract:

(xy – 1)² – (x – y)²

= x²y² – x² – y² + 1

So the original expression becomes:

(xy – 1)² – (x – y)² + 4xy

But notice:

4xy = 4xy – (–2xy + 2xy) = 4xy

We group terms differently:

x²y² – x² – y² + 1 + 4xy

= (xy + x – y + 1)(xy – x + y + 1)

Final Answer

(1 + xy + x – y)(1 + xy – x + y)