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You have provided a polynomial equation \( f(x) = x^4 - 3x^3 - 5x^2 + 9x - 2 \) and mentioned that one root of this equation is \( 2 + \sqrt{3} \). Given the rational root theorem, the conjugate \( 2 - \sqrt{3} \) should also be a root of the polynomial. I will verify this and proceed to find the remaining roots of the equation.

Let's first confirm that \( 2 + \sqrt{3} \) and \( 2 - \sqrt{3} \) are indeed roots of the polynomial by substituting these values into the polynomial. Then, I will factor the polynomial to find the other roots.

Solve f(x) = x^4 - 3x^3 - 5x^2 + 9x - 2 = 0, one root being 2 + √3.

This problem involves solving the polynomial equation f(x) = x^4 - 3x^3 - 5x^2 + 9x - 2, with one root being 2 + √3. The solution includes applying the conjugate root theorem and factorization methods. Suitable for high school students in Grades 11-12.

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