Question

Is `x^2 + 8xy - y^2` a perfect square?

Answer 1

No, it isn't, because of the `-y^2`.

Explanation:

No, it isn't. A trinomial square is always of the form `a^2 +- 2ab + b^2`. In other words, parameter `b` must always be positive. This is because `n^2`, no matter the sign of `a` gives a positive value.
If you have `(-n)^2 = (-n)(-n) = n^2`.

It is in this way that `sqrt(-4) != -2`. This is because `(sqrt(-4))^2 != (-2)^2 -> -4 != 4`. I'll also use this moment to say that the `sqrt` of any negative number is undefined in the real number syste. It is in cases such as these that we talk about imaginary numbers.

Hopefully this helps!