The points with position vectors $α\hat i+\hat j+\hat k, \hat i-\hat j-\hat k, \hat i+2\hat j-\hat k, \hat i +\hat j +β\hat k$ are coplanar if

$(1 - α) (1 + β) = 0$

Let P, Q, R and S be the given points with position vectors $α\hat i+\hat j+\hat k, \hat i-\hat j-\hat k, \hat i+2\hat j-\hat k$ and $\hat i +\hat j +β\hat k$ respectively. Then,

$\vec{QP} = (α -1)\hat i +2\hat j + 2\hat k, \vec{QR} = 0\hat i+3\hat j +0\hat k$

and $\vec{QS}=0\hat i+2\hat j+(β+ 1)\hat k$ are coplanar.

$∴[\vec{QP},\vec{QR},\vec{QS}]=0$

$⇒\begin{vmatrix}α -1&2&2\\0&3&0\\0&2&β+ 1\end{vmatrix}=0$

$⇒(α -1)(β+ 1)=0⇒(1 - α) (1 + β) = 0$