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### Section 1.6 : Rational Expressions

7. Perform the indicated operation in the following expression and reduce the answer to lowest terms.

$\frac{{\displaystyle \frac{3}{{x + 1}}}}{{\displaystyle \frac{{x + 4}}{{{x^2} + 11x + 10}}}}$

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This is just a division and so letâ€™s first convert it to a product.

$\frac{{\displaystyle \frac{3}{{x + 1}}}}{{\displaystyle \frac{{x + 4}}{{{x^2} + 11x + 10}}}} = \frac{3}{{x + 1}}\,\centerdot \,\frac{{{x^2} + 11x + 10}}{{x + 4}}$ Show Step 2

Now we can factor each of the second term as much as possible to get,

$\frac{{\displaystyle \frac{3}{{x + 1}}}}{{\displaystyle \frac{{x + 4}}{{{x^2} + 11x + 10}}}} = \frac{3}{{x + 1}}\,\centerdot \,\frac{{\left( {x + 1} \right)\left( {x + 10} \right)}}{{x + 4}}$ Show Step 3

Now cancel as much as possible to reduce to lowest terms and do the product.

$\frac{{\displaystyle \frac{3}{{x + 1}}}}{{\displaystyle \frac{{x + 4}}{{{x^2} + 11x + 10}}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{3\left( {x + 10} \right)}}{{x + 4}}}}$