Let's work through the given division step-by-step.
We need to simplify the expression:
[tex]\[
\frac{6x^2y^4 - 12x^4y^2}{3x}
\][/tex]
### Step 1: Factor the Numerator
First, we factor the expressions in the numerator:
[tex]\[
6x^2y^4 \text{ and } -12x^4y^2
\][/tex]
Both terms have a common factor of [tex]\( 6x^2 \)[/tex]:
[tex]\[
6x^2 y^4 - 12x^4 y^2 = 6x^2(y^4 - 2x^2 y^2)
\][/tex]
### Step 2: Simplify Inside the Parentheses
Inside the parentheses, recognize the common factor [tex]\( y^2 \)[/tex] in both terms:
[tex]\[
y^4 - 2x^2 y^2 = y^2(y^2 - 2x^2)
\][/tex]
So, the factored numerator is:
[tex]\[
6x^2 (y^2 (y^2 - 2x^2))
\][/tex]
### Step 3: Simplify the Fraction
Next, we need to simplify the fraction:
[tex]\[
\frac{6x^2 (y^2 (y^2 - 2x^2))}{3x}
\][/tex]
### Step 4: Divide Numerator by Denominator
We can divide each term in the numerator by [tex]\( 3x \)[/tex]:
[tex]\[
\frac{6x^2 y^2 (y^2 - 2x^2)}{3x} = 2x y^2 (y^2 - 2x^2)
\][/tex]
### Final Expression
Thus, the simplified form of the given expression is:
[tex]\[
2x y^2 (y^2 - 2x^2)
\][/tex]
To restate the solution clearly:
[tex]\[
\frac{6x^2y^4 - 12x^4y^2}{3x} = 2xy^2(y^2 - 2x^2)
\][/tex]
So, the correct answer is:
[tex]\[
2x y^4 - 4x^3 y^2
\][/tex]
In compact form, the simplified result is:
[tex]\[
2x y^2 (y^2 - 2x^2)
\][/tex]
