[tex](x - 2) \sqrt{ {x}^{2} + 2x } = (x + 1) \sqrt{ {x}^{2} - x} [/tex]

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To solve this equation, we can start by isolating the square roots on each side.

[ (x - 2) \sqrt{x^2 + 2x} = (x + 1) \sqrt{x^2 - x} ]

Square both sides to eliminate the square roots:

[ (x - 2)^2 (x^2 + 2x) = (x + 1)^2 (x^2 - x) ]

Simplify both sides:

[ (x^2 - 4x + 4)(x^2 + 2x) = (x^2 + 2x + x + 1)(x^2 - x) ]

[ (x^4 - 2x^3 - 8x^2 + 16x) = (x^3 + 3x^2 - x)(x^2 - x) ]

Expand both sides:

[ x^4 - 2x^3 - 8x^2 + 16x = x^5 - x^4 + 3x^4 - 3x^3 - x^3 + 3x^2 - x^2 + 1x^3 - x^2 ]

Combine like terms:

[ x^4 - 2x^3 - 8x^2 + 16x = x^5 + 2x^4 - 4x^3 - 4x^2 ]

Rearrange the terms and set the equation equal to zero:

[ x^5 + 2x^4 - 4x^3 - 4x^2 - x^4 + 2x^3 + 8x^2 - 16x = 0 ]

Combine like terms:

[ x^5 + x^4 - 2x^3 + 4x^2 - 16x = 0 ]

Since this is a polynomial equation of degree 5, it may be difficult to find the exact values of x that satisfy the equation. You may need to use numerical methods or other techniques to approximate the solutions.