Define a function $E({\bf w})=\beta({\bf t}-{\bf \Phi} {\bf w})^{\text T}({\bf t}-{\bf \Phi} {\bf w})+\alpha {\bf w}^{\text T}{\bf w}$, we want to prove the following equality
$E({\bf w})-E({\bf v})=({\bf w}-{\bf v})^{\text T}{\bf S}({\bf w}-{\bf v})$ where ${\bf S} = \alpha {\bf I}+\beta {\bf \Phi}^{\text T} {\bf \Phi}$ and ${\bf v} = \beta {\bf S}^{-1}{\bf \Phi}^{\text T}{\bf t}$ and ${\bf \Phi}$ is symmetric.
I have difficulty showing the equation is true by simply expanding $E$ and ${\bf S}$ (see below, ${\bf v}$ has an inverse, so I keep ${\bf v}$ in the expansion in my attempt). For example, in the LHS, terms with $\alpha$ are $\alpha({\bf w}^{\text T}{\bf w}-2{\bf w}^{\text T}{\bf v} + {\bf v}^{\text T}{\bf v})$, and the terms with $\alpha$ in the RHS is $\alpha({\bf w}^{\text T}{\bf w}- {\bf v}^{\text T}{\bf v})$. There is no way for them to equal each other in general.
Expand the LHS:
$\alpha{\bf w}^{\text T}{\bf w} - 2\beta{\bf w}^{\text T}{\bf \Phi}^{\text T}{\bf t} + \beta{\bf w}^{\text T}{\bf \Phi}^{\text T}{\bf \Phi}{\bf w} + 2\beta{\bf v}^{\text T}{\bf \Phi}^{\text T}{\bf t} - \beta{\bf v}^{\text T}{\bf \Phi}^{\text T}{\bf \Phi}{\bf v} - \alpha{\bf v}^{\text T}{\bf v} $
Expand the RHS:
$\alpha{\bf w}^{\text T}{\bf w}-2\alpha{\bf w}^{\text T}{\bf v}+\beta{\bf w}^{\text T}{\bf \Phi}^{\text T}{\bf \Phi}{\bf w}+\beta{\bf v}^{\text T}{\bf \Phi}^{\text T}{\bf \Phi}{\bf v}-2\beta{\bf w}^{\text T}{\bf \Phi}^{\text T}{\bf \Phi}{\bf v}+\alpha{\bf v}^{\text T}{\bf v}$