Let $L\colon\mathbf{P_2}\to \mathbf{P_2}$ be given by $L[p(x)] = p(2x+1)$. We want to prove it is a linear transformation.
Trying to prove that $L[u+v] = L[u]+L[v]$
$\begin{align*} L[p(x)+q(x)] &= 2[p(x)+q(x)]+1\\ &= 2p(x)+2q(x)+1 \end{align*}$
From the other size of $L[u]+L[v]$ $\begin{align*} L[q(x)]+L[q(x)] &= 2(p(x)+1)+2(q(x)+1)\\ &=2p(x)+2q(x)+4 \end{align*}$
These do result do no match, and so $L[p(x)]=p(2x+1)$ cannot not be a linear transformation?