Multiply `(9)/(2)x^(2)y` by (x + 2y)

Answer

To solve the problem of multiplying \(\frac{9}{2}x^{2}y\) by \((x + 2y)\), we will follow these steps:

### Step 1: Distribute the term \(\frac{9}{2}x^{2}y\) to each term inside the parentheses.

We have:
\[ \frac{9}{2}x^{2}y \cdot (x + 2y) \]
This can be expanded as:
\[ \frac{9}{2}x^{2}y \cdot x + \frac{9}{2}x^{2}y \cdot 2y \]

### Step 2: Multiply \(\frac{9}{2}x^{2}y\) by the first term \(x\).

Calculating this gives:
\[ \frac{9}{2}x^{2}y \cdot x = \frac{9}{2}x^{2} \cdot x^{1} \cdot y = \frac{9}{2}x^{(2+1)}y = \frac{9}{2}x^{3}y \]

### Step 3: Multiply \(\frac{9}{2}x^{2}y\) by the second term \(2y\).

Calculating this gives:
\[ \frac{9}{2}x^{2}y \cdot 2y = \frac{9}{2} \cdot 2 \cdot x^{2} \cdot y^{1} \cdot y^{1} = 9x^{2}y^{(1+1)} = 9x^{2}y^{2} \]

### Step 4: Combine the results from Steps 2 and 3.

Now we combine the two results:
\[ \frac{9}{2}x^{3}y + 9x^{2}y^{2} \]

### Final Answer:
Thus, the final expression after multiplying \(\frac{9}{2}x^{2}y\) by \((x + 2y)\) is:
\[ \frac{9}{2}x^{3}y + 9x^{2}y^{2} \]

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