g. \( \quad(p+3)^{2} *(2 p+1) \)

To expand the expression \( (p+3)^{2} * (2p+1) \), let's start with the square term. First, expand \( (p+3)^{2} \): \[ (p+3)^{2} = p^{2} + 6p + 9 \] Now, we multiply this result by \( (2p + 1) \): \[ (p^{2} + 6p + 9)(2p + 1) \] Applying the distributive property (also known as the FOIL method for binomials): - \( p^{2} * 2p = 2p^{3} \) - \( p^{2} * 1 = p^{2} \) - \( 6p * 2p = 12p^{2} \) - \( 6p * 1 = 6p \) - \( 9 * 2p = 18p \) - \( 9 * 1 = 9 \) Now, summing them all together: \[ 2p^{3} + p^{2} + 12p^{2} + 6p + 18p + 9 \] Combine like terms: \[ 2p^{3} + (1 + 12)p^{2} + (6 + 18)p + 9 = 2p^{3} + 13p^{2} + 24p + 9 \] So, the fully expanded expression is: \[ 2p^{3} + 13p^{2} + 24p + 9 \]