To simplify the given expression [tex]\(\frac{12 x^3 y^4}{6 x^3 y^2}\)[/tex], we will follow these steps:
1. Separate the coefficients and the variables:
[tex]\[ \frac{12 x^3 y^4}{6 x^3 y^2} = \frac{12}{6} \cdot \frac{x^3}{x^3} \cdot \frac{y^4}{y^2} \][/tex]
2. Simplify the coefficients:
[tex]\[ \frac{12}{6} = 2 \][/tex]
3. Simplify the [tex]\(x\)[/tex]-terms:
[tex]\[ \frac{x^3}{x^3} = 1 \][/tex]
Since the numerator and denominator are the same, they cancel each other out.
4. Simplify the [tex]\(y\)[/tex]-terms:
[tex]\[ \frac{y^4}{y^2} = y^{4-2} = y^2 \][/tex]
When dividing exponents with the same base, you subtract the exponents.
5. Combine all simplified parts:
[tex]\[ 2 \cdot 1 \cdot y^2 = 2y^2 \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ \frac{12 x^3 y^4}{6 x^3 y^2} = 2y^2 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{2y^2} \][/tex]
