To identify the factors of the expression [tex]\(x^2 + 25y^2\)[/tex], let's examine the given options and the properties of the expression step-by-step.
### Step-by-Step Solution:
1. Recognize the Form of the Expression:
- The given expression is [tex]\(x^2 + 25y^2\)[/tex].
- Notice that this is a sum of squares where [tex]\(x^2\)[/tex] is the square of [tex]\(x\)[/tex] and [tex]\(25y^2\)[/tex] is the square of [tex]\(5y\)[/tex].
2. Check Possible Factorizations:
- Option 1: [tex]\((x + 5y)(x + 5y)\)[/tex]
- If we expand [tex]\((x + 5y)(x + 5y)\)[/tex], we get:
[tex]\[
(x + 5y)^2 = x^2 + 2 \cdot x \cdot 5y + (5y)^2 = x^2 + 10xy + 25y^2
\][/tex]
- This does not match the original expression [tex]\(x^2 + 25y^2\)[/tex], since the original expression does not have the [tex]\(10xy\)[/tex] term.
- Option 2: [tex]\((x + 5y)(x - 5y)\)[/tex]
- If we expand [tex]\((x + 5y)(x - 5y)\)[/tex], we get:
[tex]\[
(x + 5y)(x - 5y) = x^2 - (5y)^2 = x^2 - 25y^2
\][/tex]
- This does not match the original expression [tex]\(x^2 + 25y^2\)[/tex], as it results in a difference of squares, not a sum.
- Option 3: [tex]\((x - 5y)(x - 5y)\)[/tex]
- If we expand [tex]\((x - 5y)(x - 5y)\)[/tex], we get:
[tex]\[
(x - 5y)^2 = x^2 - 2 \cdot x \cdot 5y + (5y)^2 = x^2 - 10xy + 25y^2
\][/tex]
- This does not match the original expression [tex]\(x^2 + 25y^2\)[/tex], since the original expression does not have the [tex]\(-10xy\)[/tex] term.
- Given these attempts, it is clear that none of the provided factorizations yield the original expression [tex]\(x^2 + 25y^2\)[/tex].
3. Determine the Expression is Prime:
- Since none of the standard factorizations match the original expression in the given form, and we cannot factor it further into products of binomials over the real numbers, we conclude that the expression [tex]\(x^2 + 25y^2\)[/tex] cannot be factored using common algebraic methods.
4. Conclusion:
- Therefore, the correct answer is that the expression [tex]\(x^2 + 25y^2\)[/tex] is prime.
The appropriate option that identifies the correct nature of the expression is:
- [tex]\( \boxed{4} \)[/tex] Prime
