To solve the problem of multiplying the polynomials [tex]\((x+3)(3x^2 + 8x + 9)\)[/tex], we need to use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.
Given polynomials:
[tex]\[ (x+3) \][/tex]
[tex]\[ (3x^2 + 8x + 9) \][/tex]
Start by distributing each term in [tex]\((x+3)\)[/tex] to each term in [tex]\((3x^2 + 8x + 9)\)[/tex]:
1. Multiply [tex]\(x\)[/tex] by each term in [tex]\(3x^2 + 8x + 9\)[/tex]:
[tex]\[ x \cdot 3x^2 = 3x^3 \][/tex]
[tex]\[ x \cdot 8x = 8x^2 \][/tex]
[tex]\[ x \cdot 9 = 9x \][/tex]
So, distributing [tex]\(x\)[/tex]:
[tex]\[ x(3x^2 + 8x + 9) = 3x^3 + 8x^2 + 9x \][/tex]
2. Multiply [tex]\(3\)[/tex] by each term in [tex]\(3x^2 + 8x + 9\)[/tex]:
[tex]\[ 3 \cdot 3x^2 = 9x^2 \][/tex]
[tex]\[ 3 \cdot 8x = 24x \][/tex]
[tex]\[ 3 \cdot 9 = 27 \][/tex]
So, distributing [tex]\(3\)[/tex]:
[tex]\[ 3(3x^2 + 8x + 9) = 9x^2 + 24x + 27 \][/tex]
Combine all the terms from the distribution:
[tex]\[ 3x^3 + 8x^2 + 9x + 9x^2 + 24x + 27 \][/tex]
Now, combine like terms:
[tex]\[ 3x^3 + (8x^2 + 9x^2) + (9x + 24x) + 27 \][/tex]
[tex]\[ 3x^3 + 17x^2 + 33x + 27 \][/tex]
Therefore, the result of the multiplication is:
[tex]\[ 3x^3 + 17x^2 + 33x + 27 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\text{B. } 3x^3 + 17x^2 + 33x + 27} \][/tex]
