If $\mathit{a}$ and$\mathit{b}$ are non-collinear vectors, then the value of $\lambda$ for which $\mathbf{u}=(\lambda +2)\mathbf{a}+\mathbf{b}$ and $\mathbf{v}=(1+4\lambda )\mathbf{a}-2\mathbf{b}$ are collinear is

$\begin{array}{l}\mathbf{u}=t\mathbf{v}\Rightarrow (\lambda +2)\mathbf{a}+\mathbf{b}=t(1+4\lambda )\mathbf{a}-2t\mathbf{b}\\ \Rightarrow (\lambda +2-t-4t\lambda )\mathbf{a}=(-2t-1)\mathbf{b}\end{array}$

Since $\mathit{a}$ and $\mathit{b}$ are non-collinear, so 

$t=-\frac{1}{2}\text{ and }\lambda +2-t-4+\lambda =0$

$\begin{array}{l}\Rightarrow \lambda +2-\frac{1}{2}-2\lambda =0\\ \Rightarrow -\lambda +\frac{3}{2}=0\Rightarrow \lambda =\frac{3}{2}.\end{array}$