Answer: Well let's see, to simplify the expression (5x^2 - y^2)(x^2 + 3y^2) + (2x^2 + y^2)(x^2 - 3y^2), we can use the distributive property and combine like terms.
Let's start by multiplying the first two terms:
(5x^2 - y^2)(x^2 + 3y^2) = 5x^2 * x^2 + 5x^2 * 3y^2 - y^2 * x^2 - y^2 * 3y^2
Simplifying each term:
= 5x^4 + 15x^2y^2 - x^2y^2 - 3y^4
Next, let's multiply the last two terms:
(2x^2 + y^2)(x^2 - 3y^2) = 2x^2 * x^2 + 2x^2 * (-3y^2) + y^2 * x^2 - y^2 * 3y^2
Simplifying each term:
= 2x^4 - 6x^2y^2 + x^2y^2 - 3y^4
Now, we can combine the like terms:
(5x^2 - y^2)(x^2 + 3y^2) + (2x^2 + y^2)(x^2 - 3y^2) = (5x^4 + 15x^2y^2 - x^2y^2 - 3y^4) + (2x^4 - 6x^2y^2 + x^2y^2 - 3y^4)
Combining like terms:
= 5x^4 + 15x^2y^2 - x^2y^2 - 3y^4 + 2x^4 - 6x^2y^2 + x^2y^2 - 3y^4
Simplifying further:
= (5x^4 + 2x^4) + (15x^2y^2 - 6x^2y^2 - x^2y^2) + (-3y^4 - 3y^4)
= 7x^4 + 8x^2y^2 - 4x^2y^2 - 6y^4
Combining like terms once again:
= 7x^4 + (8x^2y^2 - 4x^2y^2) - 6y^4
Simplifying:
= 7x^4 + 4x^2y^2 - 6y^4
Therefore, the simplified form of the expression (5x^2 - y^2)(x^2 + 3y^2) + (2x^2 + y^2)(x^2 - 3y^2) is 7x^4 + 4x^2y^2 - 6y^4.