Answer
$\dfrac{-2x^2+10x+18}{x^4+18x^2+81}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\dfrac{(x^2+9)\cdot2-(2x-5)\cdot2x}{(x^2+9)^2}
,$ use the Distributive Property first. Then remove the grouping symbols and combine like terms. Finally, use special products to simplify the denominator.
$\bf{\text{Solution Details:}}$
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{(x^2\cdot2+9\cdot2)-(2x\cdot2x-5\cdot2x)}{(x^2+9)^2}
\\\\=
\dfrac{(2x^2+18)-(4x^2-10x)}{(x^2+9)^2}
.\end{array}
Removing the grouping symbols and then combining like terms, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{2x^2+18-4x^2+10x}{(x^2+9)^2}
\\\\=
\dfrac{-2x^2+10x+18}{(x^2+9)^2}
.\end{array}
Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{-2x^2+10x+18}{(x^2)^2+2(x^2)(9)+(9)^2}
\\\\=
\dfrac{-2x^2+10x+18}{x^4+18x^2+81}
.\end{array}
