$$ ( 2 x - y ) ^ { 2 } - ( x - \partial y ) ( x + \partial y ) $$

Question

UpStudy AI · Accepted Answer

Let's simplify the given expression step by step:

$$
(2x - y)^2 - (x - \partial y)(x + \partial y)
$$

**Step 1: Expand $$(2x - y)^2$$**

$$
(2x - y)^2 = (2x)^2 - 2 \cdot 2x \cdot y + y^2 = 4x^2 - 4xy + y^2
$$

**Step 2: Expand $$(x - \partial y)(x + \partial y)$$**

This resembles the difference of squares formula $$ (a - b)(a + b) = a^2 - b^2 $$:

$$
(x - \partial y)(x + \partial y) = x^2 - (\partial y)^2
$$

**Step 3: Subtract the second expansion from the first**

$$
(4x^2 - 4xy + y^2) - (x^2 - (\partial y)^2) = 4x^2 - 4xy + y^2 - x^2 + (\partial y)^2
$$

**Step 4: Combine like terms**

$$
4x^2 - x^2 = 3x^2
$$

So, the simplified expression is:

$$
3x^2 - 4xy + y^2 + (\partial y)^2
$$

**Final Answer:**

$$
3x^2 - 4xy + y^2 + (\partial y)^2
$$
The simplified expression is $$3x^2 - 4xy + y^2 + (\partial y)^2$$.

\( ( 2 x - y ) ^ { 2 } - ( x - \partial y ) ( x + \partial y ) \)