Multiply the polynomials and simplify: (x2y+2)(x2−y) (1 point) Responses x4y−xy+x2y x to the 4th power y minus x y plus x squared y x3y−x2y2+2x2−2y x cubed y minus x squared y squared plus 2 x squared minus 2 y x4y+x2−2y x to the 4th power y plus x squared minus 2 y x4y−x2y2+2x2−2y

Answer 1

To multiply the polynomials \((x^2y + 2)(x^2 - y)\), we can use the distributive property (also known as the FOIL method for binomials).

1. Distribute \(x^2y\) across \(x^2 - y\):

   • \(x^2y \cdot x^2 = x^4y\)
   • \(x^2y \cdot (-y) = -x^2y^2\)

2. Distribute \(2\) across \(x^2 - y\):

   • \(2 \cdot x^2 = 2x^2\)
   • \(2 \cdot (-y) = -2y\)

Next, we combine all these results: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

Thus, the result is: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

Now, let's match it to your provided options:

• \(x^4y - xy + x^2y\) (does not match)
• \(x^3y - x^2y^2 + 2x^2 - 2y\) (does not match, while \(x^3y\) term is incorrect)
• \(x^4y + x^2 - 2y\) (does not match, missing \( -x^2y^2\))
• \(x^4y - x^2y^2 + 2x^2 - 2y\) (this one matches with the derived result)

Therefore, the correct simplified expression is: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

So, the final answer is: \[ \boxed{x^4y - x^2y^2 + 2x^2 - 2y} \]