Inner product identity for Hilbert space

Question

Let $H$ be a Hilbert space over $\mathbb{R}$, $\{e_i\}_{i\in I}$ a finite orthonormal set in $H$ for a given index set $I$ and $x\in H$. Also let $\{\alpha_i\}_{i\in I}$ be a family of real numbers.

I cannot understand how to go from line $(1)$ to line $(2)$ in the following:

$$\begin{align}\tag{1}&||x||^2-2\sum\limits_{i\in I}\alpha_i\langle x | e_i\rangle + \sum\limits_{i\in I}|\alpha_i|^2 \\ \tag{2}=&||x||^2-\sum\limits_{i\in I}|\langle x|e_i\rangle|^2 + \sum\limits_{i\in I}|\alpha_i-\langle x|e_i\rangle|^2\end{align}$$

Answer 1

Since the Hilbert Space is over $\mathbb{R}$, the absolute values are all unnecessary. Grouping the terms in the sums, we have \begin{align*}(\alpha_i - \langle x | e_i \rangle)^2 - \langle x | e_i \rangle^2 &= \alpha_i^2 + \langle x | e_i \rangle ^2 - 2\alpha_i \langle x|e_i\rangle - \langle x | e_i \rangle^2 \\ &= \alpha_i^2 - 2 \alpha \langle x | e_i \rangle \end{align*}