ĐKXĐ : `x\ne-9`

`x^2+(81x^2)/(x+9)^2=40`

`⇔x^2+(18x^2)/(x+9) + (81x^2)/(x+9)^2 - (18x^2)/(x+9) -40=0`

`⇔ (x-(9x)/(x+9))^2 + (18x^2)/(x+9)-40=0`

`⇔ (x^2+9x-9x)^2/(x+9)^2 + (18x^2)/(x+9)-40=0`

`⇔ x^4/(x+9)^2 + 18 . x^2/(x+9)-40=0`

`⇔x^4/(x+9)^2 +20 . x^2/(x+9) -2. x^2/(x+9)-40=0`

`⇔x^2/(x+9) (x^2/(x+9)+20)-2(x^2/(x+9)+20)=0`

`⇔ (x^2/(x+9)+20)(x^2/(x+9)-2)=0`

`⇔` $\left[ \begin{array}{l}\dfrac{x^2}{x+9}+20=0\\\dfrac{x^2}{x+9}-2=0\end{array} \right.$ 

`⇔` $\left[ \begin{array}{l}\dfrac{x^2}{x+9}=-20\\\dfrac{x^2}{x+9}=2\end{array} \right.$ 

`⇔` $\left[ \begin{array}{l}-20(x+9)=x^2\\x^2=2(x+9)\end{array} \right.$ 

`⇔` $\left[ \begin{array}{l}x^2+20x+180=0\\x^2=2x+18\end{array} \right.$ 

`⇔` $\left[ \begin{array}{l}(x+10)^2+80=0  (1) \\x^2-2x-18=0 (2)\end{array} \right.$ 

Giải `(1):(x+10)^2+80=0`

`∀x∈RR` ta có : `(x+10)^2≥0`

`⇒(x+10)^2+80>0`

`⇒` Phương trình `(1)` vô nghiệm .

Giải `(2):x^2-2x-18=0`

`Δ=b^2-4ac`

`=(-2)^2-4.1.(-18)`

`=76>0`

`⇒` Phương trình `(2)` có hai nghiệm phân biệt 

`x_1=1+sqrt19`  `(` TMĐK `)`

`x_2=1-sqrt19`   `(` TMĐK `)`

Vậy `S={1+sqrt19 ; 1-sqrt19}`