What is the product of the polynomials (x2y+2)(x2−y)?(1 point)

Question

Responses

x3y−x2y2+2x2−2y
x cubed y minus x squared y squared plus 2 x squared minus 2 y

x4y−x2y2+2x2−2y
x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y

x4y−xy+x2y
x superscript 4 baseline y minus x y plus x squared y

x4y+x2−2y
x superscript 4 baseline y plus x squared minus 2 y

Answer 1

To find the product of 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 \(x^2 - y\): \[ x^2y \cdot x^2 + x^2y \cdot (-y) = x^4y - x^2y^2 \]

2. Distribute \(2\) across \(x^2 - y\): \[ 2 \cdot x^2 + 2 \cdot (-y) = 2x^2 - 2y \]

Now, combine all the terms from both distributions: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

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

Looking through the provided responses, the one that matches this result is: \[ \text{x}^4\text{y} - \text{x}^2\text{y}^2 + 2\text{x}^2 - 2\text{y} \]

Thus, the correct answer is: \[ \text{x}^4\text{y} - \text{x}^2\text{y}^2 + 2\text{x}^2 - 2\text{y} \]