Ali was asked to factorise x^2y^2 + 36 - 4x^2 - 9y^2. He tried some ways of grouping terms as shown below.

x^2y^2 + 36 - 4x ^2 - 9y^2 = (x^2y^2 + 36) - (4x ^2 + 9y^2)

x^2y^2 + 36 - 4x^2 - 9y^2 = (x^2y^2 + 36 - 4x^2) - 9y^2

x^2y^2 + 36 - 4x^2 - 9y^2=x^2y^2 + (36 - 4x^2 - 9y^2)

As he could not carry out factorisation with the above groupings, he concluded that the expression could not be factorised. Do you agree with him? Why or why not?

It can be factorised

Step-by-step explanation:

Given

x²y² + 36 - 4x² - 9y² (rearranging)

x²y² - 4x² - 9y² + 36 (factor first/second and third/fourth terms)

= x² (y² - 4) - 9 (y² - 4) ← factor out (y² - 4) from each term

= (y² - 4) (x² - 9)

Both factors are a difference of squares and factor in general as

a² - b² = (a - b) (a + b)

Thus

y² - 4

= y² - 2² = (y - 2) (y + 2), and

x² - 9

= x² - 3² = (x - 3) (x + 3)

Hence

x²y² + 36 - 4x² - 9y² = (y - 2) (y + 2) (x - 3) (x + 3)