1. Which of the following expressions is not equivalen to $\frac{4 x y^{\prime}+2 x^{-2} y}{2 x y^{\prime}}$ ? (1) $2+x$ (3) $\frac{1}{2 x y}\left(4 x y+2 x^{2} y\right)$ (2) $3 x^{2} y$ (4) $\frac{4 x y}{2 x y}+\frac{2 x^{2} y}{2 x y}$

Step 1 ：The original expression is \(\frac{4 x y^{\prime}+2 x^{-2} y}{2 x y^{\prime}}\). We can simplify this by dividing each term in the numerator by the denominator, which gives us \(2 + \frac{y}{x^{3}y^{\prime}}\).

Step 2 ：Now we simplify each of the given expressions and compare them to the simplified original expression. If any of the simplified given expressions is not equal to the simplified original expression, then that expression is not equivalent to the original expression.

Step 3 ：The first expression is \(2+x\), which simplifies to \(x + 2\).

Step 4 ：The second expression is \(3 x^{2} y\), which simplifies to \(3x^{2}y\).

Step 5 ：The third expression is \(\frac{1}{2 x y}\left(4 x y+2 x^{2} y\right)\), which simplifies to \(x + 2\).

Step 6 ：The fourth expression is \(\frac{4 x y}{2 x y}+\frac{2 x^{2} y}{2 x y}\), which simplifies to \(x + 2\).

Step 7 ：Comparing these simplified expressions to the simplified original expression, we see that none of them are equivalent.

Step 8 ：\(\boxed{\text{Final Answer: All of the given expressions are not equivalent to the original expression. Therefore, the expressions that are not equivalent to }\frac{4 x y^{\prime}+2 x^{-2} y}{2 x y^{\prime}}\text{ are (1) }2+x\text{, (2) }3 x^{2} y\text{, (3) }\frac{1}{2 x y}\left(4 x y+2 x^{2} y\right)\text{, and (4) }\frac{4 x y}{2 x y}+\frac{2 x^{2} y}{2 x y}\text{.}}\)