To determine the factors of the expression [tex]\( x^2 + 16y^2 \)[/tex], we need to analyze given options and see if any correctly match the expansion of the expression when factored.
Let's evaluate each of the given options:
1. Option 1: [tex]\((x + 4y)(x + 4y)\)[/tex]
Expand this option:
[tex]\[
(x + 4y)(x + 4y) = x^2 + 4yx + 4yx + 16y^2 = x^2 + 8xy + 16y^2
\][/tex]
The result is [tex]\( x^2 + 8xy + 16y^2 \)[/tex], which does not match [tex]\( x^2 + 16y^2 \)[/tex].
2. Option 2: [tex]\((x + 4y)(x - 4y)\)[/tex]
Expand this option:
[tex]\[
(x + 4y)(x - 4y) = x^2 - 4yx + 4yx - 16y^2 = x^2 - 16y^2
\][/tex]
The result is [tex]\( x^2 - 16y^2 \)[/tex], which again does not match [tex]\( x^2 + 16y^2 \)[/tex].
3. Option 3: Prime
This means that the expression cannot be factored over the set of real numbers. We will evaluate this considering real numbers only.
4. Option 4: [tex]\((x - 4y)(x - 4y)\)[/tex]
Expand this option:
[tex]\[
(x - 4y)(x - 4y) = x^2 - 4yx - 4yx + 16y^2 = x^2 - 8xy + 16y^2
\][/tex]
The result is [tex]\( x^2 - 8xy + 16y^2 \)[/tex], which does not match [tex]\( x^2 + 16y^2 \)[/tex].
After examining all the options, we see that none of them factor correctly into [tex]\( x^2 + 16y^2 \)[/tex] using real numbers. Hence, the expression [tex]\( x^2 + 16y^2 \)[/tex] is indeed prime and does not factor into simpler polynomials.
Therefore, the correct answer is:
[tex]\[
\boxed{\text{Prime}}
\][/tex]
