To determine which expression is equivalent to [tex]\( 6xy(y + 3x) - 2x(3xy + y^2) \)[/tex], we need to simplify the given expression step-by-step.
First, expand each term inside the parentheses:
[tex]\[ 6xy(y + 3x) = 6xy \cdot y + 6xy \cdot 3x \][/tex]
Simplify each part of the expanded form:
[tex]\[ 6xy \cdot y = 6xy^2 \][/tex]
[tex]\[ 6xy \cdot 3x = 18x^2y \][/tex]
So, the first part of our expression becomes:
[tex]\[ 6xy(y + 3x) = 6xy^2 + 18x^2y \][/tex]
Now, expand the second part:
[tex]\[ -2x(3xy + y^2) = -2x \cdot 3xy - 2x \cdot y^2 \][/tex]
Simplify each part of the expanded form:
[tex]\[ -2x \cdot 3xy = -6x^2y \][/tex]
[tex]\[ -2x \cdot y^2 = -2xy^2 \][/tex]
So, the second part of our expression becomes:
[tex]\[ -2x(3xy + y^2) = -6x^2y - 2xy^2 \][/tex]
Now, combine the two expanded expressions:
[tex]\[ 6xy^2 + 18x^2y - 6x^2y - 2xy^2 \][/tex]
Combine like terms:
[tex]\[ (6xy^2 - 2xy^2) + (18x^2y - 6x^2y) \][/tex]
[tex]\[ 4xy^2 + 12x^2y \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ 4xy^2 + 12x^2y \][/tex]
Thus, the correct choice is:
B) [tex]\(4xy^2 + 12x^2y\)[/tex]
