`(mx-ny)(mx-ny)=` ____

Options

A. `m^(2)x^(2)+2mxny-n^(2)y^(2)`
B. `m^(2)x^(2)-2mxny-n^(2)y^(2)`
C. `m^(2)x^(2)-2mxny+n^(2)y^(2)`
D. `m^(2)x^(2)+2mxny+n^(2)y^(2)`

Answer

To solve the expression \((mx - ny)(mx - ny)\), we can use the algebraic identity for the square of a binomial. The identity states that:

\[ (a - b)^2 = a^2 - 2ab + b^2 \]

In our case, we can identify:
- \(a = mx\)
- \(b = ny\)

Now, we can apply this identity step by step:

1. **Identify \(a\) and \(b\)**:
- Here, \(a = mx\) and \(b = ny\).

2. **Apply the identity**:
- According to the identity, we have:
\[ (mx - ny)^2 = (mx)^2 - 2(mx)(ny) + (ny)^2 \]

3. **Calculate each term**:
- The first term is \((mx)^2 = m^2x^2\).
- The second term is \(-2(mx)(ny) = -2mnyx\).
- The third term is \((ny)^2 = n^2y^2\).

4. **Combine the terms**:
- Putting it all together, we get:
\[ (mx - ny)^2 = m^2x^2 - 2mnyx + n^2y^2 \]

Thus, the final answer is:

\[ (m x - n y)(m x - n y) = m^2x^2 - 2mnyx + n^2y^2 \]