Final Answer
$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
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Step-by-step Solution
$\frac{x^2+x-2}{x^2+5x+6}$
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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 $-\frac{25}{4}$
$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
Intermediate steps
A binomial squared (difference) is equal to the square of the first term, minus the double product of the first by the second, plus the square of the second term. In other words: $(a-b)^2=a^2-2ab+b^2$
$\frac{-\frac{9}{4}+x^2+x+\frac{1}{4}}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
Add the values $-\frac{9}{4}$ and $\frac{1}{4}$
$\frac{-2+x^2+x}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
A binomial squared (difference) is equal to the square of the first term, minus the double product of the first by the second, plus the square of the second term. In other words: $(a-b)^2=a^2-2ab+b^2$
$-\frac{1}{4}+x^2+5x+\frac{25}{4}$
Add the values $-\frac{1}{4}$ and $\frac{25}{4}$
$6+x^2+5x$
$\frac{-2+x^2+x}{6+x^2+5x}$
Explain more
9
Divide $-2+x^2+x$ by $6+x^2+5x$
$\begin{array}{l}\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}{\phantom{;}1\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+5x\phantom{;}+6\overline{\smash{)}\phantom{;}x^{2}+x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}\underline{-x^{2}-5x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{-x^{2}-5x\phantom{;}-6\phantom{;}\phantom{;};}-4x\phantom{;}-8\phantom{;}\phantom{;}\\\end{array}$
$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
Final Answer$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$