Correct Answer: Option C

Explanation

To find the values of \( x \) where the curve 

\(y = x^3 + 2x^2 - 5x - 6\)

crosses the x-axis, we set \( y = 0 \):

\(x^3 + 2x^2 - 5x - 6 = 0\)

Testing for rational roots, we find that \( x = -1 \) is a root. Factoring the polynomial gives:

\(x^3 + 2x^2 - 5x - 6 = (x + 1)(x^2 + x - 6)\)

Solving the quadratic \( x^2 + x - 6 = 0 \) using the quadratic formula yields:

\(x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot -6}}{2 \cdot 1} = \frac{-1 \pm 5}{2}\)

Thus, the solutions are:

\(x = -1, \quad x = 2, \quad x = -3\)

Therefore, the values of \( x \) where the curve crosses the x-axis are  \(x = -1, \quad x = 2, \quad x = -3\).

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