Final answer:
To solve the equation x³ + 2x² + 5x + 10 = 0, divide by (x + 2) to get a quadratic factor which can then be solved.
Step-by-step explanation:
To solve the equation x³ + 2x² + 5x + 10 = 0, we can use the long division method or factor theorem to find one of the roots and then use synthetic division or long division to divide the given equation by the factor found. In this case, we can use synthetic division to divide the equation by (x + 2) as a root, which gives us (x + 2)(x² + x + 5) = 0. The quadratic factor can be solved using the quadratic formula or factoring.
The quadratic factor (x² + x + 5) does not have real solutions because the discriminant is negative. Therefore, the only real root of the original equation is x = -2.