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

Answer

To find the product of the expressions \((x^3 - y^3)(x^2 + y^2)\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Distribute \(x^3\) to both terms in the second expression
We start by multiplying \(x^3\) with each term in the expression \((x^2 + y^2)\):

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

### Step 2: Simplify the products
Now, we simplify each term:

1. \(x^3 \cdot x^2 = x^{3+2} = x^5\)
2. \(x^3 \cdot y^2 = x^3y^2\)

So, after this step, we have:

\[ x^5 + x^3y^2 \]

### Step 3: Distribute \(-y^3\) to both terms in the second expression
Next, we multiply \(-y^3\) with each term in \((x^2 + y^2)\):

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

### Step 4: Simplify the products
Now, we simplify each term:

1. \(-y^3 \cdot x^2 = -x^2y^3\)
2. \(-y^3 \cdot y^2 = -y^{3+2} = -y^5\)

So, after this step, we have:

\[ -x^2y^3 - y^5 \]

### Step 5: Combine all the terms
Now we combine all the terms we have obtained:

\[ x^5 + x^3y^2 - x^2y^3 - y^5 \]

### Final Answer
Thus, the final product is:

\[ x^5 + x^3y^2 - x^2y^3 - y^5 \]

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