Final answer to the problem
$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
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Step-by-step Solution
Specify the solving method
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 $x^2+x+\frac{1}{4}$
$\frac{\left(x+\sqrt{\left(\frac{1}{4}\right)}\right)^2-2-\frac{1}{4}}{x^2+5x+6}$
3
Subtract the values $-2$ and $-\frac{1}{4}$
$\frac{-\frac{9}{4}+\left(x+\sqrt{\left(\frac{1}{4}\right)}\right)^2}{x^2+5x+6}$
4
Calculate the square root of $\frac{1}{4}$
$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{x^2+5x+6}$
5
Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$
$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{x^2+5x+6+\frac{25}{4}-\frac{25}{4}}$
6
Factor the perfect square trinomial $x^2+5x+\frac{25}{4}$
$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{\left(x+\frac{5}{2}\right)^2+6-\frac{25}{4}}$
7
Subtract the values $6$ and $-\frac{25}{4}$
$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
8
Expand
$\frac{-2+x^2+x}{6+x^2+5x}$
9
Divide $-2+x^2+x$ by $6+x^2+5x$
$\begin{array}{l}\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}{\phantom{;}1\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+5x\phantom{;}+6\overline{\smash{)}\phantom{;}x^{2}+x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}\underline{-x^{2}-5x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{-x^{2}-5x\phantom{;}-6\phantom{;}\phantom{;};}-4x\phantom{;}-8\phantom{;}\phantom{;}\\\end{array}$
10
Resulting polynomial
$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
Final answer to the problem
$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$