If the sum and the difference of two expressions is `5x^(2)-x-4` and `x^(2)+9x-10` respectively, ten the expressions are:

Options

A. `(4x^(2)+8x-6)` and `(4x^(2)-10x+2)`
B. `(2x^(2)+4x-3)` and `(3x^(2)-10x-6)`
C. `(3x^(2)+4x-7)` and `(2x^(2)-5x+3)`
D. `(3x^(2)+4x+7)` and `(2x^(2)-5x-3)`

Answer

To solve the problem, we need to find two expressions \( P \) and \( Q \) given their sum and difference.

1. **Define the expressions**:
Let \( P \) and \( Q \) be the two expressions we need to find.

2. **Set up the equations based on the problem statement**:
- The sum of the expressions is given as:
\[ P + Q = 5x^2 - x - 4 \quad \text{(Equation 1)} \]
- The difference of the expressions is given as:
\[ P - Q = x^2 + 9x - 10 \quad \text{(Equation 2)} \]

3. **Add the two equations**:
To eliminate \( Q \), we can add Equation 1 and Equation 2:
\[ (P + Q) + (P - Q) = (5x^2 - x - 4) + (x^2 + 9x - 10) \]
Simplifying the left side:
\[ 2P = (5x^2 + x^2) + (-x + 9x) + (-4 - 10) \]
This simplifies to:
\[ 2P = 6x^2 + 8x - 14 \]

4. **Solve for \( P \)**:
Divide both sides by 2:
\[ P = 3x^2 + 4x - 7 \]

5. **Substitute \( P \) back into Equation 1 to find \( Q \)**:
Using Equation 1:
\[ P + Q = 5x^2 - x - 4 \]
Substitute \( P \):
\[ (3x^2 + 4x - 7) + Q = 5x^2 - x - 4 \]
Rearranging gives:
\[ Q = (5x^2 - x - 4) - (3x^2 + 4x - 7) \]
Simplifying:
\[ Q = 5x^2 - x - 4 - 3x^2 - 4x + 7 \]
Combine like terms:
\[ Q = (5x^2 - 3x^2) + (-x - 4x) + (-4 + 7) \]
This results in:
\[ Q = 2x^2 - 5x + 3 \]

6. **Final expressions**:
Thus, the two expressions are:
\[ P = 3x^2 + 4x - 7 \]
\[ Q = 2x^2 - 5x + 3 \]