Answer
$\displaystyle \frac{1}{x^{2}-3x+9}$
Work Step by Step
Step by step multiplication of rational expressions:
1. Factor completely what you can
2. Reduce (divide) numerators and denominators by common factors.
3. Multiply the remaining factors in the numerators and
multiply the remaining factors in the denominators. $(\displaystyle \frac{P}{Q}\cdot\frac{R}{S}=\frac{PR}{QS})$
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Factor what we can:
... recognize a square of a sum: $a^{2}+2ab+b^{2}=(a+b)^{2}$
$x^{2}+6x+9=x^{2}+2(x)(3)+3^{2}=(x+3)^{2}$
... recognize a sum of cubes, $a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$
$x^{3}+27=x^{3}+3^{3}=(x+3)(x^{2}-3x+9)$
The problem becomes
$...=\displaystyle \frac{(x+3)^{2}\cdot 1}{(x-3)(x^{2}+3x+9)\cdot(x+3)}\qquad$ ... divide out the common factors
$=\displaystyle \frac{\fbox{$(x+3)^{2}$}\cdot 1}{\fbox{$(x+3)$}(x^{2}-3x+9)\cdot\fbox{$(x+3)$}}$
= $\displaystyle \frac{1}{x^{2}-3x+9}$
