Factorise this: 25x^2 + 16y^2?

It can't be factorised in the set of real numbers, as the sum of two squares doesn't have any real roots (it is verified only when both x and y are equal to zero).

Not sure if you are meant to solve this with complex numbers.

(5x + 4iy)(5x - 4iy) is a possibility.
Or it could just as well be
(4y + 5ix)(4y - 5ix).

25x^2 + 16y^2
= 25x^2 - (-16y^2)
= (5x)^2 - (4iy)^2
= (5x - 4iy)(5x + 4iy)

Ok

(5x+4y)(5x-4y)

25x^2 + 16y^2 = (5x)^2 - (4iy)^2 = (5x-4iy)(5x+4iy).

25x² + 16y²

= (5x)² + (4y)²

= (5x)² - (-4y)²

= (5x)² - (i 4y)²

= (5x + i4y) (5x - i4y)

25x^2 + 16y^2 =
(5x + 4iy)(5x - 4iy)

cannot be factored.

Did you mean 25x^2 -16y^2?