Factor out the greatest common factor from the following polynomial. \( 18 a^{2} b^{2}-24 a^{5} b^{5}+24 a b+12 a^{5} b \)

To factor out the greatest common factor (GCF) from the polynomial \( 18 a^{2} b^{2} - 24 a^{5} b^{5} + 24 a b + 12 a^{5} b \), we first determine the GCF of the coefficients and the variables. The coefficients are 18, -24, 24, and 12. The GCF of these numbers is 6. Now for the variables: - For \( a \), the lowest power present is \( a^1 \) (from \( 24ab \)). - For \( b \), the lowest power present is \( b^1 \) (also from \( 24ab \)). Thus, the GCF of the entire polynomial is \( 6ab \). Now we factor out \( 6ab \) from each term: \[ 18 a^{2} b^{2} = 6ab \cdot 3ab \] \[ -24 a^{5} b^{5} = 6ab \cdot -4a^{4}b^{4} \] \[ 24 a b = 6ab \cdot 4 \] \[ 12 a^{5} b = 6ab \cdot 2a^{4} \] Now, we can rewrite the polynomial as: \[ 6ab(3ab - 4a^{4}b^{4} + 4 + 2a^{4}) \] Thus, the factored form of the polynomial is: \[ 6ab(3ab - 4a^{4}b^{4} + 4 + 2a^{4}) \]