Factor polyomial \( 6 g^{2}+23 g+7 \)

To factor the polynomial \( 6g^2 + 23g + 7 \), we can look for two numbers that multiply to \( 6 \times 7 = 42 \) and add up to \( 23 \). The numbers \( 21 \) and \( 2 \) fit this. Rewriting the middle term, we have: \[ 6g^2 + 21g + 2g + 7 \] Now, group the terms: \[ (6g^2 + 21g) + (2g + 7) \] Factor by grouping: \[ 3g(2g + 7) + 1(2g + 7) \] This gives us: \[ (3g + 1)(2g + 7) \] So, the factored form of the polynomial is \( (3g + 1)(2g + 7) \). If you're interested in checking your work, you can always expand the factors back out to make sure you arrive at the original polynomial again—it's a foolproof method! Plus, it’s satisfying to see how the pieces fit back together!