I have an expression: $E[(b+X)^2|Y]$ where $X$ and $Y$ are normally distributed random variables, being two components of a final unknown outcome $Z$ ($Y$ is known, $X$ is the noise component):
$Y$ = $Z$ + $X$.
Their distributions are normal and independent: $X$ ~ $N(0,\sigma_x^2)$ and $Y$ ~ $N(0,\sigma_y^2)$.
Z and X are normally distributed and independent: $X$ ~ $N(0,\sigma_x^2)$ and $Z$ ~ $N(0,\sigma_z^2)$.
$b$ is just a linear function of, among other things, $Y$ : $b = \lambda_yY+\lambda_qQ+\delta$.
Sorry, I have no idea how to solve for the expectation...