To solve this equation, we need to isolate one of the square roots and then square both sides of the equation to eliminate the square roots.

Given: [tex]\sqrt{4x+1} - \sqrt{x-2} = 3[/tex]

Move the square root of (x-2) to the other side:
[tex]\sqrt{4x+1} = 3 + \sqrt{x-2}[/tex]

Square both sides of the equation to eliminate the square roots:
tex^2 = (3 + \sqrt{x-2})^2[/tex]
[tex]4x+1 = 9 + 6\sqrt{x-2} + x - 2[/tex]

Combine like terms:
[tex]3x - 6 = 6\sqrt{x-2}[/tex]

Square both sides again to get rid of the remaining square root:
[tex](3x - 6)^2 = (6\sqrt{x-2})^2[/tex]
[tex]9x^2 - 36x + 36 = 36(x-2)[/tex]
[tex]9x^2 - 36x + 36 = 36x - 72[/tex]
[tex]9x^2 - 72x + 108 = 0[/tex]

Factor the quadratic equation:
[tex]9(x^2 - 8x + 12) = 0[/tex]
[tex]9(x-2)(x-6) = 0[/tex]

Solve for x:
[tex]x-2 = 0 \implies x = 2[/tex]
[tex]x-6 = 0 \implies x = 6[/tex]

Therefore, the solutions are x = 2, x = 6.