If ` P = m^(2) - n^(2), Q = n + m, R = 2m + 2n + m^(2),` then P + 2Q - R is equal to

Options

A. `m^(2)`
B. `-m^(2)`
C. `n^(2)`
D. `-n^(2)`

Answer

To solve the expression \( P + 2Q - R \) given the values of \( P \), \( Q \), and \( R \):

1. **Identify the values**:
- \( P = m^2 - n^2 \)
- \( Q = n + m \)
- \( R = 2m + 2n + m^2 \)

2. **Substitute the values into the expression**:
\[ P + 2Q - R = (m^2 - n^2) + 2(n + m) - (2m + 2n + m^2) \]

3. **Expand the expression**:
- First, calculate \( 2Q \):
\[ 2Q = 2(n + m) = 2n + 2m \]
- Now substitute this back into the expression:
\[ P + 2Q - R = (m^2 - n^2) + (2n + 2m) - (2m + 2n + m^2) \]

4. **Remove the brackets**:
\[ = m^2 - n^2 + 2n + 2m - 2m - 2n - m^2 \]

5. **Combine like terms**:
- The \( m^2 \) terms:
\[ m^2 - m^2 = 0 \]
- The \( n \) terms:
\[ -n^2 \quad (\text{only } -n^2 \text{ remains}) \]
- The \( m \) terms:
\[ 2m - 2m = 0 \]
- The \( n \) terms:
\[ 2n - 2n = 0 \]

6. **Final result**:
\[ P + 2Q - R = -n^2 \]

Thus, the final answer is \( -n^2 \).