Find each of the following products :
`(x^(2) - 5x + 8) xx (x^(2) + 2x - 3)`

Answer

To find the product of the algebraic expressions \( (x^2 - 5x + 8) \) and \( (x^2 + 2x - 3) \), we will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first expression by each term in the second expression.

### Step-by-Step Solution:

1. **Write down the expressions:**
\[ (x^2 - 5x + 8) \times (x^2 + 2x - 3) \]

2. **Distribute each term in the first expression:**
- Multiply \( x^2 \) by each term in the second expression:
\[ x^2 \cdot x^2 = x^4 \]
\[ x^2 \cdot 2x = 2x^3 \]
\[ x^2 \cdot (-3) = -3x^2 \]

- Multiply \( -5x \) by each term in the second expression:
\[ -5x \cdot x^2 = -5x^3 \]
\[ -5x \cdot 2x = -10x^2 \]
\[ -5x \cdot (-3) = 15x \]

- Multiply \( 8 \) by each term in the second expression:
\[ 8 \cdot x^2 = 8x^2 \]
\[ 8 \cdot 2x = 16x \]
\[ 8 \cdot (-3) = -24 \]

3. **Combine all the results:**
\[ x^4 + 2x^3 - 3x^2 - 5x^3 - 10x^2 + 15x + 8x^2 + 16x - 24 \]

4. **Combine like terms:**
- Combine \( x^3 \) terms:
\[ 2x^3 - 5x^3 = -3x^3 \]
- Combine \( x^2 \) terms:
\[ -3x^2 - 10x^2 + 8x^2 = -5x^2 \]
- Combine \( x \) terms:
\[ 15x + 16x = 31x \]

5. **Write the final expression:**
\[ x^4 - 3x^3 - 5x^2 + 31x - 24 \]

### Final Answer:
\[ x^4 - 3x^3 - 5x^2 + 31x - 24 \]