Let's carefully analyze the steps provided to determine which expression is missing in step 7.
Given:
[tex]\[
(\sqrt{1 + d^2})^2 + (\sqrt{e^2 + 1})^2
\][/tex]
Step-by-step simplification:
1. Consider each term separately:
[tex]\[
(\sqrt{1 + d^2})^2 = 1 + d^2
\][/tex]
[tex]\[
(\sqrt{e^2 + 1})^2 = e^2 + 1
\][/tex]
2. Adding these simplified results together:
[tex]\[
(1 + d^2) + (e^2 + 1)
\][/tex]
Simplifying further by combining like terms:
[tex]\[
1 + d^2 + e^2 + 1 = 2 + d^2 + e^2
\][/tex]
3. According to the problem, this should be equal to:
[tex]\[
d^2 - 2de + e^2
\][/tex]
4. Equate the two expressions:
[tex]\[
2 + d^2 + e^2 = d^2 - 2de + e^2
\][/tex]
5. To solve for the missing term, subtract [tex]\( d^2 \)[/tex] and [tex]\( e^2 \)[/tex] from both sides:
[tex]\[
2 = -2de
\][/tex]
6. Then solve for [tex]\( de \)[/tex]:
[tex]\[
-1 = de
\][/tex]
Therefore, the missing expression that equates the two sides in step 7 is:
[tex]\[
\boxed{-2 d e}
\][/tex]
