(sqrt(x + 1) + sqrt(x - 2))(x - 3sqrt(x - 2) + 2) = 9

To solve the equation \((\sqrt{x + 1} + \sqrt{x - 2})(x - 3\sqrt{x - 2} + 2) = 9\), let's start by simplifying and analyzing it step by step.

Let: \[ y_1 = \sqrt{x + 1} \] \[ y_2 = \sqrt{x - 2} \] Then, we rewrite \(x\) in terms of \(y_2\): \[ x = y_2^2 + 2 \] Substituting \(y_1\) and \(y_2\) in the original equation gives: \[ (y_1 + y_2)((y_2^2 + 2) - 3y_2 + 2) = 9 \] This simplifies to: \[ (y_1 + y_2)(y_2^2 - 3y_2 + 4) = 9 \]

Next, since \(y_1 = \sqrt{y_2^2 + 1}\), we can rewrite the equation as: \[ (\sqrt{y_2^2 + 1} + y_2)(y_2^2 - 3y_2 + 4) = 9 \]

Now we can consider solving this equation numerically or through substitution methods. However, the presence of square roots complicates the analytic approach.

Instead, we can check for rational roots by substituting some values for \(x\):

1. Testing \(x = 3\): \[ \sqrt{3 + 1} + \sqrt{3 - 2} = \sqrt{4} + \sqrt{1} = 2 + 1 = 3 \] Then, \[ 3 - 3\sqrt{3 - 2} + 2 = 3 - 3(1) + 2 = 3 - 3 + 2 = 2 \] Thus, \[ 3 \cdot 2 = 6 \quad (\text{not a solution}) \]

2. Testing \(x = 5\): \[ \sqrt{5 + 1} + \sqrt{5 - 2} = \sqrt{6} + \sqrt{3} \] Then, \[ 5 - 3\sqrt{5 - 2} + 2 = 5 - 3\sqrt{3} + 2 = 7 - 3\sqrt{3} \] This leads to: \[ (\sqrt{6} + \sqrt{3})(7 - 3\sqrt{3}) = 9 \quad (\text{requires further simplifying}) \]

3. Testing \(x = 6\): \[ \sqrt{6 + 1} + \sqrt{6 - 2} = \sqrt{7} + \sqrt{4} = \sqrt{7} + 2 \] Then, \[ 6 - 3\sqrt{6 - 2} + 2 = 6 - 3\cdot 2 + 2 = 6 - 6 + 2 = 2 \] This leads to: \[ (\sqrt{7} + 2)(2) = 2\sqrt{7} + 4 \] Check if \(2\sqrt{7} + 4 = 9\) implies \(2\sqrt{7} = 5\) which is not a solution.

Due to complexity, numerical or graphical approach might help further find roots or refine the solution areas. A graphical calculator or numerical methods can also be employed if no clear algebraic simplification comes.

Finally, we would typically reformulate our numerical attempts or numeric solver for refining results on calculations of x to fit original equation leading towards solutions, or alternatively resort to software solutions for non-analytic methods to give a solution set if regular integers aren't yielding solvable states.

Would you like to pursue that numeric or graphical approach here?