Simplify (2xy^2+3)((2xy^2)^2-(2xy^2)(3)+(3)^2)

This reminds me of

(U+a)(U^2- aU + a^2). When mulipltiplied out, one gets a cubic equation. Do you remember the formula for factoring the sum of two cubes?

a^3+b^3= ????

Yes, the formula for factoring the sum of two cubes is:

a^3 + b^3 = (a + b)(a^2 - ab + b^2).

In this case, we have (2xy^2)^3 + 3^3, which can be simplified using the formula to:

(2xy^2 + 3)((2xy^2)^2 - (2xy^2)(3) + (3)^2).

Now, let's simplify each term step by step:

(2xy^2)^2 = (2xy^2)(2xy^2) = 4x^2y^4.
(2xy^2)(3) = 6xy^2.

Substituting these values into the expression:

(2xy^2 + 3)(4x^2y^4 - 6xy^2 + 9).

And that is the simplified form of the expression (2xy^2 + 3)((2xy^2)^2 - (2xy^2)(3) + (3)^2).

Yes, I remember the formula for factoring the sum of two cubes. The formula is:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

To simplify the expression (2xy^2 + 3)((2xy^2)^2 - (2xy^2)(3) + (3)^2), let's first rewrite it in the general form of the sum of two cubes:

(a + b)(a^2 - ab + b^2)

In this case, we can let a = 2xy^2 and b = 3. So, the expression can be written as:

(2xy^2 + 3)((2xy^2)^2 - (2xy^2)(3) + (3)^2)

= (2xy^2 + 3)(4x^2y^4 - 6xy^2 + 9)

Now, we can apply the formula for factoring the sum of two cubes by substituting a = 2xy^2 and b = 3 into the formula:

(2xy^2 + 3)(4x^2y^4 - 6xy^2 + 9) = (2xy^2 + 3)((2xy^2)^2 - (2xy^2)(3) + (3)^2)

= (2xy^2 + 3)(4x^2y^4 - 6xy^2 + 9)

= (2xy^2 + 3)((2xy^2)^2 - 2xy^2(3) + 3^2)

= (2xy^2 + 3)(4x^2y^4 - 6xy^2 + 9)

Therefore, the expression (2xy^2 + 3)((2xy^2)^2 - (2xy^2)(3) + (3)^2) cannot be further simplified.