It's kind of interesting to think about the best way to explain the error.

In line 3, when the expanded terms are reduced to a binomial, neither 3-5 nor 5-4 are binomials in the usual sense of unknown variables that cannot be summed because they are unknown. The usual procedure is to consolidate the constants, which would have given us -1 squared and 1 squared in line 4. Which are indeed equal.

But then, it would have been much more obvious that square roots are commonly limited to the positive roots even though they are both positive and negative. For example, the square roots of 4 are 2 and -2, but only 2 is commonly written down. But here it is arbitrarily written, in effect, that one side the root is only -1 while on the other it is only +1. It should have been +1

*and* -1 on

*both* sides.

PS Another way of putting it: It is obvious that the square root of a^2 is both plus and minus a. But if we write (a)^2, the parentheses appear to exclude the negative root. The moral here is that parentheses are conveniences, not genuine mathematical operations.