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\frac{-5\left(3\sqrt{5}-2\right)}{\left(3\sqrt{5}+2\right)\left(3\sqrt{5}-2\right)}
Rationalize the denominator of \frac{-5}{3\sqrt{5}+2} by multiplying numerator and denominator by 3\sqrt{5}-2.
\frac{-5\left(3\sqrt{5}-2\right)}{\left(3\sqrt{5}\right)^{2}-2^{2}}
Consider \left(3\sqrt{5}+2\right)\left(3\sqrt{5}-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{-5\left(3\sqrt{5}-2\right)}{3^{2}\left(\sqrt{5}\right)^{2}-2^{2}}
Expand \left(3\sqrt{5}\right)^{2}.
\frac{-5\left(3\sqrt{5}-2\right)}{9\left(\sqrt{5}\right)^{2}-2^{2}}
Calculate 3 to the power of 2 and get 9.
\frac{-5\left(3\sqrt{5}-2\right)}{9\times 5-2^{2}}
The square of \sqrt{5} is 5.
\frac{-5\left(3\sqrt{5}-2\right)}{45-2^{2}}
Multiply 9 and 5 to get 45.
\frac{-5\left(3\sqrt{5}-2\right)}{45-4}
Calculate 2 to the power of 2 and get 4.
\frac{-5\left(3\sqrt{5}-2\right)}{41}
Subtract 4 from 45 to get 41.
\frac{-15\sqrt{5}+10}{41}
Use the distributive property to multiply -5 by 3\sqrt{5}-2.