Let
P = 12xy - 10y$^2$ – 18x$^2$,
Q = 14x$^2$ + 12y$^2$ + 9xy, and
R = 5y$^2$ - x$^2$ + xy
then (P + Q) – R =

Question

Accepted Answer

To solve the problem and verify why Option 4 is the correct answer, we need to perform the operations as specified in the question and compare the result with the given options.

**Step-by-Step Explanation:**

1. **Calculate P + Q:**

Given:
   - $ P = 12xy - 10y^2 - 18x^2 $
   - $ Q = 14x^2 + 12y^2 + 9xy $

Add P and Q:
   $$
   P + Q = (12xy - 10y^2 - 18x^2) + (14x^2 + 12y^2 + 9xy)
   $$

Combine like terms:
   - Coefficients of $ x^2 $: $-18x^2 + 14x^2 = -4x^2$
   - Coefficients of $ y^2 $: $-10y^2 + 12y^2 = 2y^2$
   - Coefficients of $ xy $: $12xy + 9xy = 21xy$

Thus, $ P + Q = 21xy - 4x^2 + 2y^2 $.

2. **Subtract R from (P + Q):**

Given:
   - $ R = 5y^2 - x^2 + xy $

Subtract R from the result of $ P + Q $:
   $$
   (P + Q) - R = (21xy - 4x^2 + 2y^2) - (5y^2 - x^2 + xy)
   $$

Distribute the negative sign and combine like terms:
   - Coefficients of $ x^2 $: $-4x^2 - (-x^2) = -4x^2 + x^2 = -3x^2$
   - Coefficients of $ y^2 $: $2y^2 - 5y^2 = -3y^2$
   - Coefficients of $ xy $: $21xy - xy = 20xy$

Thus, $(P + Q) - R = 20xy - 3x^2 - 3y^2$.

3. **Verify with the given options:**

- **Option 1:** $22xy - 3x^2 + 3y^2$ (Incorrect: Coefficients do not match)
   - **Option 2:** $22xy + 3x^2 - 3y^2$ (Incorrect: Coefficients do not match)
   - **Option 3:** $20xy - 7x^2 - 3y^2$ (Incorrect: Coefficients do not match)
   - **Option 4:** $20xy - 3x^2 - 3y^2$ (Correct: Matches exactly)

Therefore, Option 4 is the correct answer because it matches the result of the calculation $(P + Q) - R$ exactly. The other options are incorrect as their coefficients do not align with the calculated expression.