To solve this equation, start by isolating one of the square root terms on one side of the equation:

√(x+10) - √(x+3) = √(4x-23)

Add √(x+3) to both sides:

√(x+10) = √(x+3) + √(4x-23)

Now square both sides of the equation to eliminate the square root terms:

(x+10) = (x+3) + 2√((x+3)(4x-23)) + (4x-23)

Expand the terms:

x + 10 = x + 3 + 2√(4x^2 + 12x - 92) + 4x - 23

Combine like terms:

10 = -20 + 2√(4x^2 + 12x - 92)

30 = 2√(4x^2 + 12x - 92)

Divide by 2:

15 = √(4x^2 + 12x - 92)

Square both sides again to eliminate the square root:

225 = 4x^2 + 12x - 92

Rearrange the equation:

4x^2 + 12x - 92 - 225 = 0

4x^2 + 12x - 317 = 0

This is a quadratic equation that can be solved using the quadratic formula:

x = [-b ± √(b^2 - 4ac)] / 2a

For this equation, a = 4, b = 12, and c = -317.

x = [-12 ± √(12^2 - 44(-317))] / 2*4
x = [-12 ± √(144 + 5072)] / 8
x = [-12 ± √(5216)] / 8
x = [-12 ± 72] / 8

Solving for both the positive and negative roots, we get:

x = (60 / 8) = 7.5 or x = (-84 / 8) = -10.5

Therefore, the solutions to the equation are x = 7.5 and x = -10.5.