To solve this problem using the Arithmetic Mean–Geometric Mean (AM–GM) inequality, we aim to find the maximum value of $(X + 3Y)^2$ under the given constraint $(X^2) + (9Y^2) = 10$.

Step 1: Restate the Problem

We are given the equation:

$X^2 + 9Y^2 = 10$

We need to maximize the expression:

$(X + 3Y)^2$

Step 2: Apply AM–GM Inequality

The AM–GM inequality states that for any non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.

To apply this concept, notice that:

$X^2 + 9Y^2 = 10$

We can rewrite $X^2 + 9Y^2$ in terms of squares of sums, noting that this is the sum of two squares (similar to how AM-GM works):

The AM-GM inequality doesn't directly apply here as a shortcut for maximizing $(X + 3Y)^2$, but we can solve it using standard maximization techniques like Lagrange multipliers or substitution methods for now.

Step 3: Parametrize the Problem Using Trigonometry (Alternate Approach)

One effective approach is to parametrize $X^2 + 9Y^2 = 10$ by substituting trigonometric identities. We can let:

$X = \sqrt{10} \cos \theta, \quad Y = \frac{\sqrt{10}}{3} \sin \theta$

Thus, we want to maximize:

$(X + 3Y)^2 = (\sqrt{10} \cos \theta + \sqrt{10} \sin \theta)^2 = 10 (\cos \theta + \sin \theta)^2$

Using the identity $\cos \theta + \sin \theta = \sqrt{2} \cos \left( \theta - \frac{\pi}{4} \right)$, we get:

$(X + 3Y)^2 = 10 \times 2 = 20$

Thus, the maximum value of $(X + 3Y)^2$ is $20$.

Conclusion:

The maximum value of $(X + 3Y)^2$ is $\boxed{20}$.

Would you like further clarification on any steps? Here are some related questions you might find helpful:

1. How do trigonometric parametrizations work in maximizing constrained expressions?
2. What is the AM–GM inequality, and how can it be applied in different cases?
3. How would you solve a constrained optimization problem using Lagrange multipliers?
4. What other inequalities are useful in mathematical optimization problems?
5. How does the identity $\cos \theta + \sin \theta = \sqrt{2} \cos \left( \theta - \frac{\pi}{4} \right)$ help in solving such problems?

Tip: When maximizing a quadratic expression under a constraint, consider using trigonometric substitution for a cleaner solution.