LEN-Model equivalency
The main point is that the principal's expected utility from a payoff $z$ conditional on a certain level of effort $e$ can be written as

$$ \text{E}[{z}|e] - \frac{r_p}{2}\text{Var}(z|e).$$

In other words, since wealth is normally distributed, exponential utility has a simple `mean-variance' representation. For a derivation, see here.

I take it that the principal's payoff $z$ equals $x - w(x) = (1 - b)x - a$. It is then straightforward to compute the (conditional) mean and variance of $z$:

$$ \text{E}[z|e] = (1 - b)\text{E}[x|e] - \text{E}[a] = (1 - b)\mu(e) - a,$$

$$\text{Var}[z|e] = (1 - b)^2 \text{Var}(x|e) - \text{Var}(a)= (1 - b)^2\sigma^2. $$

It follows that the principal's expected utility can be written as

$$ (1 - b)\mu(e) - a - \frac{r_p}{2}(1 - b)^2\sigma^2.$$