Simplify (3/x-x/3)((3x)/(x^2+6x+9))

Rewrite ${\displaystyle 9}$ as ${\displaystyle 3^2}$.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

Rewrite the polynomial.

Factor using the perfect square trinomial rule ${\displaystyle a^2+2ab+b^2={(a+b)}^2}$, where ${\displaystyle a=x}$ and ${\displaystyle b=3}$.

To write ${\displaystyle \frac{3}{x}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{3}{3}}$.

To write ${\displaystyle -\frac{x}{3}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{x}{x}}$.

Write each expression with a common denominator of ${\displaystyle x\cdot 3}$, by multiplying each by an appropriate factor of ${\displaystyle 1}$.

Combine the numerators over the common denominator.

Simplify the numerator.

Combine exponents.

Reduce the expression ${\displaystyle \frac{(3+x)(3-x)\cdot 3x}{3x}}$ by cancelling the common factors.

Cancel the common factor of ${\displaystyle x}$.

Expand ${\displaystyle (3+x)(3-x)}$ using the FOIL Method.

Simplify and combine like terms.

Rewrite ${\displaystyle 9}$ as ${\displaystyle 3^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=3}$ and ${\displaystyle b=x}$.

Reorder terms.

Cancel the common factors.