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The roots of the cubic equation x^3 + 2x^2 + x + 1 = 0 are to be found.

To find the roots of a cubic equation, we can use the Rational Root Theorem to identify possible rational roots. The Rational Root Theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must have a numerator that divides the constant term and a denominator that divides the leading coefficient.

In this case, the constant term is 1 and the leading coefficient is 1, so the possible rational roots are ±1. We can test these roots by substituting them into the equation and checking if they satisfy it.

Substituting x = -1, we get (-1)^3 + 2(-1)^2 - 1 + 1 = -1 + 2 - 1 + 1 = 1, which is not equal to 0. Therefore, x = -1 is not a root.

Substituting x = 1, we get 1^3 + 2(1)^2 + 1 + 1 = 5, which is not equal to 0. Therefore, x = 1 is not a root.

Since neither ±1 is a root, we can conclude that there are no rational roots. However, we can still find the roots using numerical methods such as Newton-Raphson or the cubic formula. The cubic formula is quite complicated, so we will use a calculator to find the approximate roots:

x ≈ -1.5321, -0.3473 + 0.5906i, -0.3473 - 0.5906i

Therefore, the roots of the cubic equation x^3 + 2x^2 + x + 1 = 0 are approximately -1.5321, -0.3473 + 0.5906i, and -0.3473 - 0.5906i.

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