Find each of the following products :
`(2x^(2) + 3x - 7) xx (3x^(2) - 5x + 4)`

Answer

To find the product of the two algebraic expressions \( (2x^2 + 3x - 7) \) and \( (3x^2 - 5x + 4) \), we will use the distributive property (also known as the FOIL method for binomials). Here are the steps to solve the problem:

### Step-by-Step Solution

1. **Write down the expressions**:
\[ (2x^2 + 3x - 7) \times (3x^2 - 5x + 4) \]

2. **Distribute each term in the first expression to each term in the second expression**:
- Multiply \( 2x^2 \) by each term in \( (3x^2 - 5x + 4) \):
\[ 2x^2 \cdot 3x^2 = 6x^4 \]
\[ 2x^2 \cdot (-5x) = -10x^3 \]
\[ 2x^2 \cdot 4 = 8x^2 \]

- Multiply \( 3x \) by each term in \( (3x^2 - 5x + 4) \):
\[ 3x \cdot 3x^2 = 9x^3 \]
\[ 3x \cdot (-5x) = -15x^2 \]
\[ 3x \cdot 4 = 12x \]

- Multiply \( -7 \) by each term in \( (3x^2 - 5x + 4) \):
\[ -7 \cdot 3x^2 = -21x^2 \]
\[ -7 \cdot (-5x) = 35x \]
\[ -7 \cdot 4 = -28 \]

3. **Combine all the results**:
Now we combine all the terms we calculated:
\[ 6x^4 + (-10x^3 + 9x^3) + (8x^2 - 15x^2 - 21x^2) + (12x + 35x) - 28 \]

4. **Simplify the expression**:
- Combine like terms:
- For \( x^3 \): \( -10x^3 + 9x^3 = -x^3 \)
- For \( x^2 \): \( 8x^2 - 15x^2 - 21x^2 = -28x^2 \)
- For \( x \): \( 12x + 35x = 47x \)

Putting it all together, we have:
\[ 6x^4 - x^3 - 28x^2 + 47x - 28 \]

### Final Answer
The product of the two expressions is:
\[ 6x^4 - x^3 - 28x^2 + 47x - 28 \]