Solve lambdax^2+4lambdax+1>0

$$\lambda x ^ { 2 } + 4 \lambda x + 1 > 0$$

Solve for x

$$\left\{\begin{matrix}x\neq -2\text{, }&\lambda =\frac{1}{4}\\x\in \left(-\infty,-\frac{\sqrt{\lambda \left(4\lambda -1\right)}+2\lambda }{\lambda }\right)\cup \left(\frac{\sqrt{\lambda \left(4\lambda -1\right)}-2\lambda }{\lambda },\infty\right)\text{, }&\lambda >\frac{1}{4}\\x\in \left(\frac{\sqrt{\lambda \left(4\lambda -1\right)}-2\lambda }{\lambda },-\frac{\sqrt{\lambda \left(4\lambda -1\right)}+2\lambda }{\lambda }\right)\text{, }&\lambda <0\\x\in \mathrm{R}\text{, }&\lambda \geq 0\text{ and }\lambda <\frac{1}{4}\end{matrix}\right.$$

Solve for λ

$$\left\{\begin{matrix}\lambda >-\frac{1}{x\left(x+4\right)}\text{, }&x<-4\text{ or }x>0\\\lambda \in \mathrm{R}\text{, }&x=-4\text{ or }x=0\\\lambda <-\frac{1}{x\left(x+4\right)}\text{, }&x>-4\text{ and }x<0\end{matrix}\right.$$