6) \( 12 a^{2}+23 a b+10 b^{2} \)

To factor the quadratic expression \( 12a^{2} + 23ab + 10b^{2} \), we can first look for two numbers that multiply to the product of the leading coefficient \( 12 \) and the constant \( 10 \) (which is \( 120 \)), and add up to the middle coefficient \( 23 \). Those numbers are \( 20 \) and \( 3 \). Now we can split the middle term: \[ 12a^{2} + 20ab + 3ab + 10b^{2} \] Next, we group the terms: \[ (12a^{2} + 20ab) + (3ab + 10b^{2}) \] From the first group, we factor out \( 4a \): \[ 4a(3a + 5b) \] From the second group, we factor out \( b \): \[ b(3a + 10b) \] Now we can combine these: \[ (4a + b)(3a + 10b) \] So, the completely factored form of the expression is: \[ (4a + b)(3a + 10b) \]