If 5x$^2$ - 8y - 20x + 2xy = (x + a) (bx + 2y), then the value of (a – b) is:

Question

Accepted Answer

To solve the problem and justify why Option 3 is the correct answer, we need to expand and compare the given expression with the factored form.

The given expression is:

$$ 5x^2 - 8y - 20x + 2xy $$

We need to express this in the form:

$$ (x + a)(bx + 2y) $$

Expanding the right-hand side, we get:

$$ (x + a)(bx + 2y) = x(bx + 2y) + a(bx + 2y) $$
$$ = bx^2 + 2xy + abx + 2ay $$

Now, let's compare the coefficients from both expressions:

1. **Coefficient of $x^2$:**  
   From the given expression: 5  
   From the expanded form: b  
   Therefore, $ b = 5 $.

2. **Coefficient of $xy$:**  
   From the given expression: 2  
   From the expanded form: 2  
   This matches, so no new information is needed here.

3. **Coefficient of $x$:**  
   From the given expression: -20  
   From the expanded form: ab  
   Therefore, $ ab = -20 $.

4. **Coefficient of $y$:**  
   From the given expression: -8  
   From the expanded form: 2a  
   Therefore, $ 2a = -8 $  
   Solving for $ a $, we get $ a = -4 $.

Now, we have $ a = -4 $ and $ b = 5 $.

To find $ a - b $:

$$ a - b = -4 - 5 = -9 $$

Thus, the value of $ a - b $ is -9, which corresponds to Option 3.

**Explanation for why other options are incorrect:**

1. **Option 1 (9):**  
   This would imply $ a - b = 9 $, which contradicts our calculation of $ a - b = -9 $.

2. **Option 2 (-1):**  
   This would imply $ a - b = -1 $, which does not match our derived value of $ a - b = -9 $.

3. **Option 4 (1):**  
   This would imply $ a - b = 1 $, which is inconsistent with our calculation of $ a - b = -9 $.

Therefore, the correct answer is indeed Option 3, as supported by the calculations and logical reasoning provided.