Find each of the following products :
`(2x^(2) - 5y^(2)) xx (x^(2) + 3y^(2))`

Answer

To find the product of the algebraic expressions \( (2x^2 - 5y^2) \times (x^2 + 3y^2) \), we will use the distributive property (also known as the FOIL method for binomials). Here are the steps to solve the problem:

### Step 1: Distribute \( 2x^2 \) to both terms in the second bracket
Multiply \( 2x^2 \) by \( x^2 \) and \( 3y^2 \):
\[ 2x^2 \times x^2 + 2x^2 \times 3y^2 \]
Calculating these gives:
\[ 2x^4 + 6x^2y^2 \]

### Step 2: Distribute \( -5y^2 \) to both terms in the second bracket
Now, multiply \( -5y^2 \) by \( x^2 \) and \( 3y^2 \):
\[ -5y^2 \times x^2 + (-5y^2) \times 3y^2 \]
Calculating these gives:
\[ -5x^2y^2 - 15y^4 \]

### Step 3: Combine all the terms
Now, we combine all the terms we found in Steps 1 and 2:
\[ 2x^4 + 6x^2y^2 - 5x^2y^2 - 15y^4 \]

### Step 4: Simplify the expression
Combine like terms:
\[ 2x^4 + (6x^2y^2 - 5x^2y^2) - 15y^4 \]
This simplifies to:
\[ 2x^4 + 1x^2y^2 - 15y^4 \]
or simply:
\[ 2x^4 + x^2y^2 - 15y^4 \]

### Final Answer:
The product of \( (2x^2 - 5y^2) \times (x^2 + 3y^2) \) is:
\[ 2x^4 + x^2y^2 - 15y^4 \]

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