$\left\{\begin{matrix}v=\frac{4a^{2}+10a+9}{x\left(2a+3\right)^{2}}\text{, }&a\neq -\frac{3}{2}\text{ and }x\neq 0\\v\in \mathrm{C}\text{, }&\left(a=\frac{-5+\sqrt{11}i}{4}\text{ or }a=\frac{-\sqrt{11}i-5}{4}\right)\text{ and }x=0\end{matrix}\right.$

$v=\frac{4a^{2}+10a+9}{x\left(2a+3\right)^{2}}$
$a\neq -\frac{3}{2}\text{ and }x\neq 0$

$\left\{\begin{matrix}a=\frac{-6vx+\sqrt{12vx-11}+5}{4\left(vx-1\right)}\text{; }a=\frac{-6vx-\sqrt{12vx-11}+5}{4\left(vx-1\right)}\text{, }&v=0\text{ or }x\neq \frac{1}{v}\\a=-\frac{9\left(vx-1\right)}{2\left(6vx-5\right)}\text{, }&v\neq 0\text{ and }x=\frac{1}{v}\end{matrix}\right.$

$\left\{\begin{matrix}a=\frac{-6vx+\sqrt{12vx-11}+5}{4\left(vx-1\right)}\text{; }a=\frac{-6vx-\sqrt{12vx-11}+5}{4\left(vx-1\right)}\text{, }&\left(x\neq \frac{1}{v}\text{ and }x\geq \frac{11}{12v}\text{ and }v>0\right)\text{ or }\left(x=\frac{11}{12v}\text{ and }v\neq 0\right)\text{ or }\left(x\neq \frac{1}{v}\text{ and }v<0\text{ and }x\leq \frac{11}{12v}\right)\\a=-\frac{9\left(vx-1\right)}{2\left(6vx-5\right)}\text{, }&v\neq 0\text{ and }x=\frac{1}{v}\end{matrix}\right.$