x³ + 2x² + x + 2 = 0

Factor x² out of the first two terms on the left, and
factor 1 out of the last two terms on the left:

x²(x + 2) + 1(x + 2) = 0

(x + 2)(x² + 1) = 0

Use the zero factor principle:

x + 2 = 0          x² + 1 = 0
    x = -2             x² = -1_
                         x = ±Ö1
                         x = ±i

The roots are -2, i and -i

Edwin