(5/2)^2 is taken away because it is added in the (x +5/2)^2

You see, the incomplete square is (x^2 +5x).

To make that a perfect square, the square must be completed.

x^2 +5x +(5/2)^2

= (x +5/2)^2

So the (5/2)^2 was added to "complete the square".

Then,

(x +5/2)^2 +3

is not the same anymore as x^2 +5x +3 ----the original expression.

Because

(x +5/2)^2 +3

= x^2 +5x +(5/2)^2 +3.

There is a (5/2)^2 that was added.

So to maintain or to make the value of the new expression to equal the value of the original expression, another (5/2)^2 must be taken away from the new expression.

Hence,

x^2 +5x +3 = (x +5/2)^2 -(5/2)^2 +3