To solve the problem of factoring the expression [tex]\( 5a^2 + b \)[/tex] completely, let's analyze each of the proposed factorizations step-by-step.
1. Prime
Stating that an expression is "Prime" means that it cannot be factored further using integer coefficients. This is an option we keep in mind as we test the other factorizations.
2. [tex]\(a(5a + b)\)[/tex]
Let's check this:
- If we expand [tex]\( a(5a + b) \)[/tex], we get:
[tex]\[
a(5a + b) = 5a^2 + ab
\][/tex]
- This result does not match our original expression [tex]\( 5a^2 + b \)[/tex], as we have an extra [tex]\( ab \)[/tex] term, making this incorrect.
3. [tex]\(b(5a^2)\)[/tex]
Let's analyze this:
- If we expand [tex]\( b(5a^2) \)[/tex], we get:
[tex]\[
b(5a^2) = 5a^2b
\][/tex]
- Again, this result does not match our original expression [tex]\( 5a^2 + b \)[/tex], as the [tex]\( b \)[/tex] term is incorrectly multiplied, making this option incorrect.
4. [tex]\(ab(5a + 1)\)[/tex]
Let's test this option:
- If we expand [tex]\( ab(5a + 1) \)[/tex], we get:
[tex]\[
ab(5a + 1) = 5a^2b + ab
\][/tex]
- This result does not match our original expression [tex]\( 5a^2 + b \)[/tex], as it introduces additional terms involving [tex]\( ab \)[/tex], making this option incorrect.
Given that none of the proposed factorizations correctly factorize [tex]\( 5a^2 + b \)[/tex], and based on the information that the expression does not factor further into simpler expressions with integer coefficients, we conclude that:
The expression [tex]\( 5a^2 + b \)[/tex] is already in its simplest form and cannot be factored further. It is, therefore, prime.
So, the correct answer is:
[tex]\[
\boxed{\text{Prime}}
\][/tex]
