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The image has been displayed. Let's go through the mathematical operation step by step:

We need to simplify the given expression using **least common multiple (LCM)** and simplifying to the maximum. The expression is:

\[
\left( \frac{\frac{12}{1 \times 2} - \frac{2 \times 2}{5}}{\frac{"22"}{11} \times \frac{15}{1 \times 2}} \right)^2
\]

### Step-by-step simplification:

#### Step 1: Simplify the numerator
The numerator is:

\[
\frac{12}{1 \times 2} - \frac{2 \times 2}{5} = \frac{12}{2} - \frac{4}{5}
\]

\[
\frac{12}{2} = 6
\]

Now, we have:

\[
6 - \frac{4}{5}
\]

To subtract these, we need a common denominator (LCM of 1 and 5, which is 5):

\[
6 - \frac{4}{5} = \frac{30}{5} - \frac{4}{5} = \frac{26}{5}
\]

So, the simplified numerator is:

\[
\frac{26}{5}
\]

#### Step 2: Simplify the denominator
The denominator is:

\[
\frac{"22"}{11} \times \frac{15}{1 \times 2}
\]

Simplify:

\[
\frac{22}{11} = 2
\]
and
\[
\frac{15}{2}
\]

So, the denominator becomes:

\[
2 \times \frac{15}{2} = 15
\]

#### Step 3: Form the new expression
Now the expression becomes:

\[
\left( \frac{\frac{26}{5}}{15} \right)^2
\]

#### Step 4: Simplify the fraction
We can express this as:

\[
\frac{26}{5} \div 15 = \frac{26}{5} \times \frac{1}{15} = \frac{26}{75}
\]

So the expression is now:

\[
\left( \frac{26}{75} \right)^2
\]

#### Step 5: Square the fraction
Square both the numerator and denominator:

\[
\left( \frac{26}{75} \right)^2 = \frac{26^2}{75^2} = \frac{676}{5625}
\]

Thus, the final simplified result is:

\[
\frac{676}{5625}
\]

### Final Answer:
\[
\boxed{\frac{676}{5625}}
\]

Let me know if you want more details or if you have any questions!

---

Here are 5 related questions to deepen your understanding:
1. How do you find the least common multiple (LCM) of two numbers?
2. What happens if you square a fraction? 
3. How do you simplify a complex fraction like the one in the problem?
4. Can you explain how to subtract fractions with different denominators?
5. How does multiplying fractions work in this context?

**Tip**: When simplifying fractions, always check if the numerator and denominator have common factors you can cancel out for quicker results.

Resuelve la siguiente operación usando m.c.m. y simplificando al máximo la expresión:
((12 / (1 * 2) - 2 * 2 / 5) / ("22" / 11 * 15 / (1 * 2)))^2

Step-by-step solution for simplifying a complex fractional expression using the least common multiple (LCM) and squaring. This exercise helps deepen your understanding of LCM, fractions, and expression simplification. Suitable for Grades 8-10.

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