$(a)$ In $\mathbb{R}.$ Suppose $\lambda_1 x^2+\lambda_2 x\left|x\right|=0.$ Then for $x=1$ and $x=-1$ we get

$\lambda_1+\lambda_2=0$ and $\lambda_1-\lambda_2=0$ which implies $\lambda_1=\lambda_2=0.$

$(b)$ In $\mathbb{R}^+,$ $x\left|x\right|=x^2$ so ...

$(c)$ In $\mathbb{R}^-,$ $x\left|x\right|=-x^2$ so ...