Final answer to the problem
$1+\frac{-4x-8}{\left(x+\frac{5}{2}\right)^2-\frac{1}{4}}$
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Step-by-step Solution
1
Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$
$\frac{x^2+x-2+\frac{1}{4}-\frac{1}{4}}{x^2+5x+6}$
2
Factor the perfect square trinomial $x^2+x+\frac{1}{4}$
$\frac{\left(x+\sqrt{\frac{1}{4}}\right)^2-2-\frac{1}{4}}{x^2+5x+6}$
3
Calculate the power $\sqrt{\frac{1}{4}}$
$\frac{\left(x+\frac{1}{2}\right)^2-2-\frac{1}{4}}{x^2+5x+6}$
Intermediate steps
4
Simplify the addition $\left(x+\frac{1}{2}\right)^2-2-\frac{1}{4}$
$\frac{\left(x+\frac{1}{2}\right)^2-\frac{9}{4}}{x^2+5x+6}$
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Intermediate steps
5
Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$
$\frac{\left(x+\frac{1}{2}\right)^2-\frac{9}{4}}{x^2+5x+6+\frac{25}{4}-\frac{25}{4}}$
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Intermediate steps
6
Factor the perfect square trinomial $x^2+5+\frac{25}{4}$
$\frac{\left(x+\frac{1}{2}\right)^2-\frac{9}{4}}{\left(x+\frac{5}{2}\right)^2+6-\frac{25}{4}}$
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Intermediate steps
7
Simplify the addition $\left(x+\frac{5}{2}\right)^2+6-\frac{25}{4}$
$\frac{\left(x+\frac{1}{2}\right)^2-\frac{9}{4}}{\left(x+\frac{5}{2}\right)^2-\frac{1}{4}}$
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Intermediate steps
$\frac{x^2+x-2}{x^2+5x+6}$
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9
Divide $x^2+x-2$ by $x^2+5x+6$
$\begin{array}{l}\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}{\phantom{;}1\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+5x\phantom{;}+6\overline{\smash{)}\phantom{;}x^{2}+x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}\underline{-x^{2}-5x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{-x^{2}-5x\phantom{;}-6\phantom{;}\phantom{;};}-4x\phantom{;}-8\phantom{;}\phantom{;}\\\end{array}$
$1+\frac{-4x-8}{\left(x+\frac{5}{2}\right)^2-\frac{1}{4}}$
Final answer to the problem
$1+\frac{-4x-8}{\left(x+\frac{5}{2}\right)^2-\frac{1}{4}}$