Verified answer

If 2i is a root, then -2i is another root, since complex roots come in conjugate pairs that way. Use the factor theorem to convert the two known roots into factors: (ᵡ - 2i) (ᵡ + 2i)

Multiply the factors together:

ᵡ² + 4

Divide by the product of the known factors:

(-10ᵡᵌ-3ᵡ²-40ᵡ-12) / (ᵡ² + 4) = -10ᵡ - 3

Use the zero product principle on the reduced linear term:

-10ᵡ - 3 = 0

ᵡ = -3/10

So altogether the three roots of -10ᵡᵌ-3ᵡ²-40ᵡ-12 are:

ᵡ = -3/10, 2i, and -2i