Multiply. \[ \left(3 x^{3}+3\right)\left(6 x^{2}+7 x+9\right) \]

To multiply the two expressions, we'll use the distributive property (also known as the FOIL method for binomials) to ensure each term in the first expression is multiplied by each term in the second expression. Let’s distribute \( (3x^3 + 3) \) across \( (6x^2 + 7x + 9) \): 1. Multiply \( 3x^3 \) by each term in \( (6x^2 + 7x + 9) \): - \( 3x^3 \cdot 6x^2 = 18x^5 \) - \( 3x^3 \cdot 7x = 21x^4 \) - \( 3x^3 \cdot 9 = 27x^3 \) 2. Now multiply \( 3 \) by each term in \( (6x^2 + 7x + 9) \): - \( 3 \cdot 6x^2 = 18x^2 \) - \( 3 \cdot 7x = 21x \) - \( 3 \cdot 9 = 27 \) Now, combine all these results together: \[ 18x^5 + 21x^4 + 27x^3 + 18x^2 + 21x + 27 \] Thus, the final result of multiplying \( \left(3 x^{3}+3\right)\left(6 x^{2}+7 x+9\right) \) is: \[ \boxed{18x^5 + 21x^4 + 27x^3 + 18x^2 + 21x + 27} \]