`(2/(5)ab+c)(2/(5)ab-c)=` ____

Options

A. `4/(25)a^(2)b^(2)-4/(5)abc+c^(2)`
B. `4/(25)a^(2)b^(2)+4/(5)abc+c^(2)`
C. `4/(25)a^(2)b^(2)-c^(2)`
D. `4/(25)a^(2)b^(2)+c^(2)`

Answer

To solve the expression \((\frac{2}{5}ab + c)(\frac{2}{5}ab - c)\), we can use the algebraic identity for the difference of squares, which states:

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

### Step-by-step Solution:

1. **Identify \(a\) and \(b\)**:
- Here, we can identify:
- \(a = \frac{2}{5}ab\)
- \(b = c\)

2. **Apply the identity**:
- According to the difference of squares identity:
\[ (\frac{2}{5}ab + c)(\frac{2}{5}ab - c) = a^2 - b^2 \]
- Substitute \(a\) and \(b\) into the identity:
\[ = \left(\frac{2}{5}ab\right)^2 - c^2 \]

3. **Calculate \(a^2\)**:
- Calculate \(\left(\frac{2}{5}ab\right)^2\):
\[ = \frac{2^2}{5^2}(ab)^2 = \frac{4}{25}a^2b^2 \]

4. **Combine the results**:
- Now, substitute back into the equation:
\[ = \frac{4}{25}a^2b^2 - c^2 \]

5. **Final Result**:
- Therefore, the final expression is:
\[ \frac{4}{25}a^2b^2 - c^2 \]

### Final Answer:
\[ \frac{4}{25}a^2b^2 - c^2 \]