Multiply `(3x - (4)/(5)y^(2)x)` by `(1)/(2)xy`.

Answer

To solve the problem of multiplying the expression \((3x - \frac{4}{5}y^{2}x)\) by \(\frac{1}{2}xy\), we will follow these steps:

### Step 1: Distribute \(\frac{1}{2}xy\) to each term in the expression
We will multiply \(\frac{1}{2}xy\) with each term inside the parentheses.

\[ \frac{1}{2}xy \cdot (3x) - \frac{1}{2}xy \cdot \left(\frac{4}{5}y^{2}x\right) \]

### Step 2: Multiply the first term
Now we will multiply \(\frac{1}{2}xy\) with \(3x\).

\[ \frac{1}{2} \cdot 3 \cdot x \cdot x = \frac{3}{2}x^{2}y \]

### Step 3: Multiply the second term
Next, we will multiply \(\frac{1}{2}xy\) with \(-\frac{4}{5}y^{2}x\).

\[ -\frac{1}{2} \cdot \frac{4}{5} \cdot y^{2} \cdot x \cdot y = -\frac{4}{10}xy^{3} = -\frac{2}{5}xy^{3} \]

### Step 4: Combine the results
Now we combine the results from Step 2 and Step 3.

\[ \frac{3}{2}x^{2}y - \frac{2}{5}xy^{3} \]

### Final Answer
The final result of multiplying \((3x - \frac{4}{5}y^{2}x)\) by \(\frac{1}{2}xy\) is:

\[ \frac{3}{2}x^{2}y - \frac{2}{5}xy^{3} \]

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