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

Answer

To find the product of the algebraic expressions \( (3x + 2y - 4) \) and \( (x - y + 2) \), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Distribute the first term
Multiply \( 3x \) by each term in the second expression \( (x - y + 2) \):
\[ 3x \cdot x = 3x^2 \]
\[ 3x \cdot (-y) = -3xy \]
\[ 3x \cdot 2 = 6x \]

### Step 2: Distribute the second term
Now, multiply \( 2y \) by each term in the second expression \( (x - y + 2) \):
\[ 2y \cdot x = 2xy \]
\[ 2y \cdot (-y) = -2y^2 \]
\[ 2y \cdot 2 = 4y \]

### Step 3: Distribute the third term
Next, multiply \( -4 \) by each term in the second expression \( (x - y + 2) \):
\[ -4 \cdot x = -4x \]
\[ -4 \cdot (-y) = 4y \]
\[ -4 \cdot 2 = -8 \]

### Step 4: Combine all the results
Now, combine all the terms obtained from the previous steps:
\[ 3x^2 - 3xy + 6x + 2xy - 2y^2 + 4y - 4x + 4y - 8 \]

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

Putting it all together, we get:
\[ 3x^2 - xy + 2x - 2y^2 + 8y - 8 \]

### Final Result
Thus, the product of \( (3x + 2y - 4) \) and \( (x - y + 2) \) is:
\[ \boxed{3x^2 - xy + 2x - 2y^2 + 8y - 8} \]