Recursive Models of dynamic linear economics (Hansen / Sargent, 2014) - optimal linear regulator problem / solution of bellman equation p. 34 ff

The optimal linear regulator problem according to Hansen/Sargent, 2014, Recursive models of dynamic linear economies, on page 34 ff. is stated as follows:

$-Esum_{t=0}^{infty}beta^t[x_t’ R x_t+u_t’Qu_t+2u_t’Wx_t],; 0<beta<1$

subject to

$x_{t+1}=Ax_t+Bu_t+Cw_{t+1},tgeq 0$

${x_t}_{tgeq 0}$ are state variables,
${u_t}_{tgeq 0}$ are control variables, and
${w_t}_{tgeq 0}$ are stochastic innovations into the system.

The Bellman equation is

$V(x_t)=text{max}_{u_t}{-(x_t’ R x_t+u_t’Qu_t+2u_t’Wx_t)+beta E_tV(x_{t+1})}$

In the text on page 35 we have as a solution

$V(x_t)=-x_t’Px_t-rho$

with the following expressions after using an iteration model

$P=R+beta A’PA-(beta A’PB+W)(Q+beta B’PB)^{-1}(beta B’PA+W’)$

and

$rho=frac{beta}{1-beta}text{trace}{PCC’}$

I am looking for an explanation how to derive $V,P,rho$. Also the decision rule for the control variable

$u_t=-(Q+beta B’PB)^{-1}(beta B’PA+W’)x_t$

is of interest and requires clarification.