Multiply :
`2x^(2) - 4x + 5` by `x^(2) + 3x - 7`

Answer

To multiply the expressions \(2x^2 - 4x + 5\) and \(x^2 + 3x - 7\), we will use the distributive property (also known as the FOIL method for binomials). Here’s the step-by-step solution:

### Step 1: Distribute each term in the first expression to each term in the second expression.

We will multiply each term in \(2x^2 - 4x + 5\) by each term in \(x^2 + 3x - 7\).

1. **Multiply \(2x^2\) by each term in \(x^2 + 3x - 7\)**:
- \(2x^2 \cdot x^2 = 2x^4\)
- \(2x^2 \cdot 3x = 6x^3\)
- \(2x^2 \cdot (-7) = -14x^2\)

2. **Multiply \(-4x\) by each term in \(x^2 + 3x - 7\)**:
- \(-4x \cdot x^2 = -4x^3\)
- \(-4x \cdot 3x = -12x^2\)
- \(-4x \cdot (-7) = 28x\)

3. **Multiply \(5\) by each term in \(x^2 + 3x - 7\)**:
- \(5 \cdot x^2 = 5x^2\)
- \(5 \cdot 3x = 15x\)
- \(5 \cdot (-7) = -35\)

### Step 2: Combine all the results from the multiplications.

Now, we will combine all the terms we calculated:

\[ 2x^4 + 6x^3 - 14x^2 - 4x^3 - 12x^2 + 28x + 5x^2 + 15x - 35 \]

### Step 3: Combine like terms.

Now we will combine the like terms:

- **For \(x^4\)**:
- \(2x^4\)

- **For \(x^3\)**:
- \(6x^3 - 4x^3 = 2x^3\)

- **For \(x^2\)**:
- \(-14x^2 - 12x^2 + 5x^2 = -21x^2\)

- **For \(x\)**:
- \(28x + 15x = 43x\)

- **Constant term**:
- \(-35\)

### Final Result:

Combining all these, we get the final expression:

\[ 2x^4 + 2x^3 - 21x^2 + 43x - 35 \]

### Summary of the Solution:

The product of the expressions \(2x^2 - 4x + 5\) and \(x^2 + 3x - 7\) is:

\[ \boxed{2x^4 + 2x^3 - 21x^2 + 43x - 35} \]