To perform the indicated operation and simplify the result for the given expression [tex]\( 3 x^2 \left(\frac{3 x - 6}{x} \right) \)[/tex]:
1. Rewrite the Fraction Inside the Parentheses:
[tex]\[
\frac{3 x - 6}{x}
\][/tex]
2. Simplify the Fraction:
- Perform the division inside the fraction by separating the terms in the numerator:
[tex]\[
\frac{3 x - 6}{x} = \frac{3 x}{x} - \frac{6}{x} = 3 - \frac{6}{x}
\][/tex]
3. Multiply the Expression by [tex]\(3 x^2\)[/tex]:
[tex]\[
3 x^2 \left( 3 - \frac{6}{x} \right)
\][/tex]
4. Distribute [tex]\(3 x^2\)[/tex] into Each Term Inside the Parentheses:
- Calculate [tex]\(3 x^2 \cdot 3\)[/tex]:
[tex]\[
3 x^2 \cdot 3 = 9 x^2
\][/tex]
- Calculate [tex]\(3 x^2 \cdot \left( -\frac{6}{x} \right)\)[/tex]:
[tex]\[
3 x^2 \cdot \left( -\frac{6}{x} \right) = 3 x^2 \cdot -\frac{6}{x} = -18 x
\][/tex]
- Hence:
[tex]\[
3 x^2 \left( \frac{3 x - 6}{x} \right) = 9 x^2 - 18 x
\][/tex]
5. Final Simplified Expression:
[tex]\[
3 x^2 \left( \frac{3 x - 6}{x} \right) = 9 x (x - 2)
\][/tex]
Thus, the simplified result of the operation is:
[tex]\[
9 x (x - 2)
\][/tex]
