To rewrite the expression \( 24a^5b + 6ab^2 \) using a common factor, we first need to identify the greatest common factor (GCF) of the two terms.
- The first term is \( 24a^5b \).
- The second term is \( 6ab^2 \).
The coefficients \( 24 \) and \( 6 \) have a GCF of \( 6 \). For the variable part, the GCF of \( a^5 \) and \( a \) is \( a \), and for \( b \) and \( b^2 \), the GCF is \( b \).
Thus, the overall GCF of \( 24a^5b \) and \( 6ab^2 \) is \( 6ab \).
Now we can factor out \( 6ab \) from both terms:
\[ 24a^5b + 6ab^2 = 6ab(4a^4 + b) \]
Now, let's look at the options provided:
- \( 6ab(2a^5b + ab^2) \)
- \( 3ab(8a^4 + 2b) \)
- \( 6ab(12a^4 + 2ab^2) \)
- \( 8a^5b(4 + 3ab^2) \)
None of these match our factorization directly. However, we can check the third option, \( 6ab(12a^4 + 2ab^2) \):
We can rewrite it:
\[ 6ab(12a^4 + 2ab^2) = 6ab(6(2a^4) + 2ab^2) = 6ab(2(6a^4 + ab^2)) \]
This is not equivalent to our original expression, so it is not the correct rewritten form.
By checking the closest option \( 6ab(4a^4 + b) \) isn't listed, but it matches our factorization.
Therefore, while none of the given options match exactly with \( 6ab(4a^4 + b) \), the correct factorization of the expression \( 24a^5b + 6ab^2 \) is:
\[ 6ab(4a^4 + b) \]
If we must pick an option from the list given, we'd likely have to choose the closest one, keeping in mind accurate factorization isn't reflected here. If there are no other instruction or preferences, none of the provided options are correct.
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