Let A = 3x$^2$−8x+11, B = -2x$^2$+12x, C = −4x$^2$+7 and D = x$^2$−x−11.
If (A+B+C)−D = px$^2$+qx+ r, then the value of (p+q+r) is :

Question

Accepted Answer

To solve the problem and verify why Option 4 is the correct answer, we need to perform the algebraic operations as described in the question.

First, let's calculate $ A + B + C $:

- $ A = 3x^2 - 8x + 11 $
- $ B = -2x^2 + 12x $
- $ C = -4x^2 + 7 $

Adding these together:

$$
A + B + C = (3x^2 - 8x + 11) + (-2x^2 + 12x) + (-4x^2 + 7)
$$

Combine like terms:

- For $ x^2 $ terms: $ 3x^2 - 2x^2 - 4x^2 = -3x^2 $
- For $ x $ terms: $ -8x + 12x = 4x $
- For constant terms: $ 11 + 7 = 18 $

Thus, $ A + B + C = -3x^2 + 4x + 18 $.

Next, we need to subtract $ D $ from this result:

- $ D = x^2 - x - 11 $

Subtract $ D $ from $ A + B + C $:

$$
(A + B + C) - D = (-3x^2 + 4x + 18) - (x^2 - x - 11)
$$

Distribute the negative sign and combine like terms:

- For $ x^2 $ terms: $ -3x^2 - x^2 = -4x^2 $
- For $ x $ terms: $ 4x + x = 5x $
- For constant terms: $ 18 + 11 = 29 $

Thus, $ (A + B + C) - D = -4x^2 + 5x + 29 $.

Now, we have the expression in the form $ px^2 + qx + r $, where $ p = -4 $, $ q = 5 $, and $ r = 29 $.

Calculate $ p + q + r $:

$$
p + q + r = -4 + 5 + 29 = 30
$$

Therefore, the value of $ p + q + r $ is 30, which corresponds to Option 4.

**Explanation of Incorrect Options:**

1. **Option 1 (31):** This option is incorrect because the calculated sum $ p + q + r = 30 $, not 31.
2. **Option 2 (27):** This option is incorrect because the calculated sum $ p + q + r = 30 $, not 27.
3. **Option 3 (28):** This option is incorrect because the calculated sum $ p + q + r = 30 $, not 28.

In conclusion, the correct answer is indeed Option 4, as the calculations clearly show that $ p + q + r = 30 $.