Find each of the following products :
`(x^(3) - 5x^(2) + 3x + 1) xx (x^(2) - 3)`

Answer

To find the product of the two algebraic expressions \( (x^3 - 5x^2 + 3x + 1) \) and \( (x^2 - 3) \), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Write Down the Expressions
We start with the two expressions:
\[ (x^3 - 5x^2 + 3x + 1) \times (x^2 - 3) \]

### Step 2: Distribute Each Term
We will distribute each term in the first expression to each term in the second expression.

1. **Multiply \( x^3 \) by each term in \( (x^2 - 3) \)**:
\[ x^3 \cdot x^2 = x^{5} \]
\[ x^3 \cdot (-3) = -3x^3 \]

2. **Multiply \( -5x^2 \) by each term in \( (x^2 - 3) \)**:
\[ -5x^2 \cdot x^2 = -5x^{4} \]
\[ -5x^2 \cdot (-3) = 15x^2 \]

3. **Multiply \( 3x \) by each term in \( (x^2 - 3) \)**:
\[ 3x \cdot x^2 = 3x^{3} \]
\[ 3x \cdot (-3) = -9x \]

4. **Multiply \( 1 \) by each term in \( (x^2 - 3) \)**:
\[ 1 \cdot x^2 = x^2 \]
\[ 1 \cdot (-3) = -3 \]

### Step 3: Combine All the Terms
Now we will combine all the terms we obtained from the distribution:
\[ x^5 - 3x^3 - 5x^4 + 15x^2 + 3x^3 - 9x - 3 + x^2 \]

### Step 4: Combine Like Terms
Now, we will combine the like terms:
- The \( x^5 \) term: \( x^5 \)
- The \( x^4 \) term: \( -5x^4 \)
- The \( x^3 \) terms: \( -3x^3 + 3x^3 = 0 \) (these cancel out)
- The \( x^2 \) terms: \( 15x^2 + x^2 = 16x^2 \)
- The \( x \) term: \( -9x \)
- The constant term: \( -3 \)

Putting it all together, we get:
\[ x^5 - 5x^4 + 16x^2 - 9x - 3 \]

### Final Answer
The product of the two expressions is:
\[ \boxed{x^5 - 5x^4 + 16x^2 - 9x - 3} \]