To factor the given expression \( 6x^2y - 3xy - 24xy^2 + 12y^2 \):
1. Identify the Greatest Common Factor (GCF):
- The terms in the expression are \( 6x^2y \), \( -3xy \), \( -24xy^2 \), and \( 12y^2 \).
- We need to find the GCF of these terms.
- First, look at the numerical coefficients: 6, -3, -24, and 12. The greatest common divisor of these numbers is 3.
- Next, consider the variables: \( x^2y \), \( xy \), \( xy^2 \), and \( y^2 \). The factor that is common to all variables is \( y \).
Therefore, the GCF of the entire expression is \( 3y \).
2. Factor out the GCF:
- We rewrite each term by factoring out \( 3y \):
[tex]\[
\begin{aligned}
6x^2y &= 3y \cdot 2x^2, \\
-3xy &= 3y \cdot (-x), \\
-24xy^2 &= 3y \cdot (-8xy), \\
12y^2 &= 3y \cdot 4y.
\end{aligned}
\][/tex]
3. Rewrite the expression using the GCF:
- After factoring out the GCF \( 3y \), the rewritten expression is:
[tex]\[
3y(2x^2 - x - 8xy + 4y)
\][/tex]
So, the expression \( 6x^2y - 3xy - 24xy^2 + 12y^2 \) factored by taking out the greatest common factor \( 3y \) is:
[tex]\[
3y(2x^2 - x - 8xy + 4y)
\][/tex]
