Question 1209874
Let's solve this system of equations.

Given:
1.  x + y = √(4z + 3)
2.  y + z = √(4x - 1)
3.  z + x = √(4y + 5)

First, let's square each equation:

1.  (x + y)² = 4z + 3
2.  (y + z)² = 4x - 1
3.  (z + x)² = 4y + 5

Now, let's sum up all three equations:

(x + y)² + (y + z)² + (z + x)² = 4z + 3 + 4x - 1 + 4y + 5
x² + 2xy + y² + y² + 2yz + z² + z² + 2xz + x² = 4x + 4y + 4z + 7
2x² + 2y² + 2z² + 2xy + 2yz + 2xz = 4x + 4y + 4z + 7
2(x² + y² + z² + xy + yz + xz) = 4(x + y + z) + 7

Now, let's assume x = y = z. Then:

2(3x² + 3x²) = 4(3x) + 7
2(6x²) = 12x + 7
12x² = 12x + 7
12x² - 12x - 7 = 0

Using the quadratic formula, x = (12 ± √(144 + 4 * 12 * 7)) / 24
x = (12 ± √(144 + 336)) / 24
x = (12 ± √480) / 24
x = (12 ± 4√30) / 24
x = (3 ± √30) / 6

Let's test if x = y = z.

If x = y = z, then:

x + x = √(4x + 3)
2x = √(4x + 3)
4x² = 4x + 3
4x² - 4x - 3 = 0
(2x - 3)(2x + 1) = 0
x = 3/2 or x = -1/2

y + x = √(4x - 1)
2x = √(4x - 1)
4x² = 4x - 1
4x² - 4x + 1 = 0
(2x - 1)² = 0
x = 1/2

z + x = √(4x + 5)
2x = √(4x + 5)
4x² = 4x + 5
4x² - 4x - 5 = 0

Since the values of x are different for each equation when we assume x=y=z, x,y,z are not equal.

Let's subtract equation (2) from (1):

(x + y)² - (y + z)² = 4z + 3 - (4x - 1)
x² + 2xy + y² - y² - 2yz - z² = 4z + 3 - 4x + 1
x² - z² + 2y(x - z) = 4z - 4x + 4
(x - z)(x + z) + 2y(x - z) = 4(z - x)
(x - z)(x + z + 2y) = -4(x - z)

If x ≠ z, then:
x + z + 2y = -4

Similarly, we can subtract equation (3) from (2):

(y + z)² - (z + x)² = 4x - 1 - (4y + 5)
y² - x² + 2z(y - x) = 4x - 4y - 6
(y - x)(y + x) + 2z(y - x) = -4(y - x) - 6

If y ≠ x, then:
y + x + 2z = -4 - 6/(y-x)

And subtract equation (1) from (3):

(z + x)² - (x + y)² = 4y + 5 - (4z + 3)
z² - y² + 2x(z - y) = 4y - 4z + 2
(z - y)(z + y) + 2x(z - y) = -4(z - y) + 2

If z ≠ y, then:
z + y + 2x = -4 + 2/(z-y)

From the first result: x + z + 2y = -4
x + y + z + y = -4

Let's make a guess and check. Let x=3/2, then 4x-1 = 5, and sqrt(5) = y+z.
If x=3/2, then 4x-1=5, therefore y+z = sqrt(5).
If z=1/2, then 4z+3 = 5, therefore x+y=sqrt(5)
If y=1, then 4y+5 = 9, therefore x+z = 3

x+y+z = x+sqrt(5)

x+y = sqrt(5)
x+z=3
y+z = sqrt(5)

x+y+z = 4

Final Answer: The final answer is $\boxed{4}$