Find each of the following products :
`(x^(2) - 5x + 8) xx (x^(2) + 2)`

Answer

To find the product of the expressions \( (x^2 - 5x + 8) \) and \( (x^2 + 2) \), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Write the expression
We start with the expression:
\[ (x^2 - 5x + 8)(x^2 + 2) \]

### Step 2: Distribute \( x^2 \)
First, we will multiply \( x^2 \) by each term in the second expression:
\[ x^2 \cdot (x^2 + 2) = x^2 \cdot x^2 + x^2 \cdot 2 = x^4 + 2x^2 \]

### Step 3: Distribute \( -5x \)
Next, we will multiply \( -5x \) by each term in the second expression:
\[ -5x \cdot (x^2 + 2) = -5x \cdot x^2 + (-5x) \cdot 2 = -5x^3 - 10x \]

### Step 4: Distribute \( 8 \)
Now, we will multiply \( 8 \) by each term in the second expression:
\[ 8 \cdot (x^2 + 2) = 8 \cdot x^2 + 8 \cdot 2 = 8x^2 + 16 \]

### Step 5: Combine all the results
Now we combine all the terms we obtained from the distributions:
\[ x^4 + 2x^2 - 5x^3 - 10x + 8x^2 + 16 \]

### Step 6: Combine like terms
Now we will combine the like terms:
- The \( x^4 \) term: \( x^4 \)
- The \( x^3 \) term: \( -5x^3 \)
- The \( x^2 \) terms: \( 2x^2 + 8x^2 = 10x^2 \)
- The \( x \) term: \( -10x \)
- The constant term: \( 16 \)

Putting it all together, we have:
\[ x^4 - 5x^3 + 10x^2 - 10x + 16 \]

### Final Answer
Thus, the product of \( (x^2 - 5x + 8) \) and \( (x^2 + 2) \) is:
\[ \boxed{x^4 - 5x^3 + 10x^2 - 10x + 16} \]