Sqrt(x+5) + sqrt(x-3) = 4

We'll multiply, both sides of the equation, by the adjoint
expression of the left side.

[sqrt (x+5) + sqrt
(x-3)]*[sqrt (x+5) - sqrt (x-3)]= 4*[sqrt (x+5) - sqrt
(x-3)]

We'll transform the product from the left side in
the difference of squares.

[sqrt (x+5)]^2 - [sqrt (x-3)]^2=
4*[sqrt (x+5) - sqrt (x-3)]

(x+5) - (x-3)= 4*[sqrt (x+5) -
sqrt (x-3)]

We'll remove the paranthesis from the left
side:

x + 5  - x + 3 = 4*[sqrt (x+5) - sqrt
(x-3)]

We'll eliminate like
terms:

8 = 4*[sqrt (x+5) - sqrt
(x-3)]

We'll divide by 4:

2=
[sqrt (x+5) - sqrt (x-3)]

We'll add this result to the
initial equation:

sqrt(x+5) + sqrt(x-3) + sqrt (x+5) - sqrt
(x-3) = 6

We'll eliminate like
terms:

2sqrt(x+5) = 6

We'll
divide by 2:

sqrt(x+5) =
3

We'll raise to square both
sides:

[sqrt(x+5)]^2 =
3^2

x+5= 9

We'll subtract 5
both sides:

x=
9-5

x=
4

We'll verify and we'll get x = 4 as valid
solution.