# Multiply and simplify (x^2+2x-3/x^2+1x-20)/(x-1/x+5)

Question
Multiply and simplify
$$\displaystyle\frac{{{x}^{{2}}+{2}{x}-\frac{{3}}{{x}^{{2}}}+{1}{x}-{20}}}{{{x}-\frac{{1}}{{x}}+{5}}}$$

2021-02-15
Firstly, we will factorise the quadratics by splitting middle terms
$$\displaystyle{x}^{{2}}+{2}{x}-{3}={x}^{{2}}+{3}{x}-{x}-{3}={x}{\left({x}+{3}\right)}-{1}{\left({x}+{3}\right)}={\left({x}-{1}\right)}{\left({x}+{3}\right)}$$
$$\displaystyle{x}^{{2}}+{x}-{20}={x}^{{2}}+{5}{x}-{4}{x}-{20}={x}{\left({x}+{5}\right)}-{4}{\left({x}+{5}\right)}={\left({x}-{4}\right)}{\left({x}+{5}\right)}$$
Now we will replace the quadratic by their factors and reciprocate the lower fraction.
$$\displaystyle\frac{{\frac{{{x}^{{2}}+{2}{x}-{3}}}{{{x}{2}+{x}-{20}}}}}{{\frac{{{x}-{1}}}{{{x}+{5}}}}}=\frac{{{\left({x}-{1}\right)}{\left({x}+{3}\right)}}}{{{\left({x}-{4}\right)}{\left({x}+{5}\right)}}}\cdot\frac{{{x}+{5}}}{{{x}-{1}}}=\frac{{{x}+{3}}}{{{x}-{4}}}$$
Ans:$$\displaystyle\frac{{{x}+{3}}}{{{x}-{4}}}$$

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