What is the completely factored form of x⁴y + 4x²y + 5y?

To factor the expression x⁴y + 4x²y + 5y, we first look for common factors in all the terms. Here, we can see that each term has a common factor of y.

Factoring out the common factor, we get:

y (x4 + 4x2 + 5)

Next, we focus on factoring the polynomial x⁴ + 4x² + 5. Notice that this is a quadratic in form if we let u = x². So the expression becomes:

u2 + 4u + 5

To determine if this quadratic can be factored further, we can calculate the discriminant:

D = b2 - 4ac = 42 - 4(1)(5) = 16 - 20 = -4

Since the discriminant is negative, it means there are no real roots, and thus, it cannot be factored further over the real numbers. Therefore, the completely factored form of the original expression is:

y (x4 + 4x2 + 5)

This is the simplest form we can express the polynomial as it cannot be broken down into simpler factors with real coefficients.