dentify the factors of x2 + 36y2. Prime (x + 6y)(x − 6y) (x + 6y)(x + 6y) (x − 6y)(x − 6y)

To factor the expression $x^2 + 36y^2$, we recognize that it can be expressed in the form of a sum of squares. The given options to consider are:

1. $(x + 6y)(x - 6y)$
2. $(x + 6y)(x + 6y)$
3. $(x - 6y)(x - 6y)$

Step 1: Break Down the Problem

The expression $x^2 + 36y^2$ can be analyzed to see if it can be factored as a sum of squares.

Step 2: Relevant Concepts

For an expression of the form $a^2 + b^2$, it generally cannot be factored over the real numbers. However, it can often be expressed using complex numbers or in specific contexts, like using a different set of numbers.

• Recognizing $36y^2 = (6y)^2$, we have: $x^2 + (6y)^2$

Step 3: Analysis and Detail

Since the expression $x^2 + (6y)^2$ cannot be factored using real numbers into simple linear factors, we need to analyze its structure. Thus, we find that:

• The expression does not correspond to $(x + 6y)(x - 6y)$, which factors into $x^2 - (6y)^2$ (difference of squares).
• Therefore, it is not factored as pairs of $(x + 6y)(x + 6y)$ or $(x - 6y)(x - 6y)$.

Step 4: Verify and Summarize

The factors listed in the problem are incorrect for $x^2 + 36y^2$ as it is not expressible in such a form over real numbers.

Final Answer

The expression $x^2 + 36y^2$ cannot be factored further over the real numbers. It is in its simplest form.