Multiply `(3x^(2) + y^(2))` by `(2x^(2) + 3y^(2))`

Answer

To solve the problem of multiplying the expressions \( (3x^2 + y^2) \) and \( (2x^2 + 3y^2) \), we can follow these steps:

### Step 1: Distribute the first term of the first expression
We start by multiplying the first term of the first expression \( 3x^2 \) by each term in the second expression \( (2x^2 + 3y^2) \).

\[ 3x^2 \cdot 2x^2 + 3x^2 \cdot 3y^2 \]

### Step 2: Calculate the products
Now, we calculate each product:

1. \( 3x^2 \cdot 2x^2 = 6x^{2+2} = 6x^4 \)
2. \( 3x^2 \cdot 3y^2 = 9x^2y^2 \)

So, from this step, we get:
\[ 6x^4 + 9x^2y^2 \]

### Step 3: Distribute the second term of the first expression
Next, we multiply the second term of the first expression \( y^2 \) by each term in the second expression \( (2x^2 + 3y^2) \).

\[ y^2 \cdot 2x^2 + y^2 \cdot 3y^2 \]

### Step 4: Calculate the products
Now, we calculate these products:

1. \( y^2 \cdot 2x^2 = 2x^2y^2 \)
2. \( y^2 \cdot 3y^2 = 3y^{2+2} = 3y^4 \)

So, from this step, we get:
\[ 2x^2y^2 + 3y^4 \]

### Step 5: Combine all the results
Now we combine all the results from Step 2 and Step 4:

\[ 6x^4 + 9x^2y^2 + 2x^2y^2 + 3y^4 \]

### Step 6: Combine like terms
Next, we combine the like terms \( 9x^2y^2 \) and \( 2x^2y^2 \):

\[ 9x^2y^2 + 2x^2y^2 = 11x^2y^2 \]

### Final Result
Putting it all together, we get the final expression:

\[ 6x^4 + 11x^2y^2 + 3y^4 \]