SOLVE: �ã(4x+ 5) + 1 = 0

Question

�ã(4x+ 5)= -1
(�ã(4x+ 5))squared= (-1)squared
4x + 5 = 1
I know I solve for x
I get that x = 1
but my book states that x = -1
Please show me how this is true

I don't understand what the term �ã means. Can you retype the problem some other way?

Your third line suggests to me that the original question was
√(4x+5) + 1 = 0

from your "4x + 5 = 1 "

4x = 1 - 5
4x=-4
x=-1 as your book states.

HOWEVER, as soon as you reached your second line of
√(4x+5) = -1 you could have stopped, since that contradicts the definition of the square root symbol, which is defined as the positive square root of a number.

eg. suppose you had x^2 = 9

Many students are taught to have as their next line
x= ± 3 whereas the actual should have been ± x = 3

For equations it really does not matter but in inequations such as
x^2 > 9 it is actually WRONG to say
x > ± 3 and you must have ± x > 3

Secondly, once you square both sides of an equation like you did in your third line, all answers you obtain must be verified in the original equation.

So in this case the correct answer would be
"no solution in the real number set"