Certainly! Let's factor the given expression step-by-step:
Given expression:
[tex]\[12 x^3 y + 6 x^2 y^2 - 9 x y^3\][/tex]
Step 1: Identify the Greatest Common Factor (GCF) of the terms in the expression.
Each term in the expression has the factors:
1. [tex]\(12 x^3 y\)[/tex]: has factors [tex]\(3, 4, x^3, y\)[/tex]
2. [tex]\(6 x^2 y^2\)[/tex]: has factors [tex]\(3, 2, x^2, y^2\)[/tex]
3. [tex]\(9 x y^3\)[/tex]: has factors [tex]\(3, 3, x, y^3\)[/tex]
The common factors are [tex]\(3, x, y\)[/tex].
Combining these, the GCF of the three terms is:
[tex]\[3xy\][/tex]
Step 2: Factor out the GCF from each term in the expression:
[tex]\[
12 x^3 y + 6 x^2 y^2 - 9 x y^3 = 3xy \left( \frac{12 x^3 y}{3xy} + \frac{6 x^2 y^2}{3xy} - \frac{9 x y^3}{3xy} \right)
\][/tex]
Step 3: Perform the division inside the parentheses:
[tex]\[
3xy \left(4 x^2 + 2 x y - 3 y^2\right)
\][/tex]
So, the factored form of the expression is:
[tex]\[3xy \left(4 x^2 + 2 x y - 3 y^2\right)\][/tex]
Therefore, the correct factorization of the given expression [tex]\(12 x^3 y + 6 x^2 y^2 - 9 x y^3\)[/tex] is:
[tex]\[3xy \left(4 x^2 + 2 x y - 3 y^2\right)\][/tex]
Among the given options, the correct answer is:
[tex]\[3 x y\left(4 x^2+2 x y-3 y^2\right)\][/tex]
