Given subspace (of $\mathbb{R}^4$)
$V= \rm span ([2,3,1,2], [3,2,2,3], [1,-1,1,1]) $
For $\beta_1=[1,1,1,1], \beta _2=[2,-1,1,2]$ desribe set of all vectors $[b_1, b_2] \in \mathbb{R}^2 $ such that $b_1\beta_1+b_2\beta_2 \in V$ What is the most general method for that type of questions?
If you reduce the matrix $\begin{bmatrix}1&2&2&3&1\\1&-1&3&2&-1\\1&1&1&2&1\\1&2&2&3&1\end{bmatrix}$ to RREF,
you obtain $\begin{bmatrix}1&0&0&1&1\\0&1&0&\frac{1}{2}&\frac{1}{2}\\0&0&1&\frac{1}{2}&-\frac{1}{2}\\0&0&0&0&0\end{bmatrix}$; so a general vector in the nullspace is given by
$\;\;\;(b_1,b_2,a_1,a_2,a_3)=s(-2,-1,-1,2,0)+t(-2,-1,1,0,2)$ for $s,t\in\mathbb{R}$.
Therefore $[b_1,b_2]=[-2s-2t,-s-t]$ for $s,t\in\mathbb{R}$; so $[b_1,b_2]=c[2,1]$ for any $c\in\mathbb{R}$.
(Geometrically, this is the line $y=\frac{1}{2}x$ in the xy-plane.)