To solve the question, we need to determine which expression is missing from step 7. Let’s walk through the calculations step-by-step and identify the missing expression.
1. We start with the expression given in step 7:
[tex]\[
\left(\sqrt{1 + d^2}\right)^2 + \left(\sqrt{e^2 + 1}\right)^2
\][/tex]
2. Squaring each term:
[tex]\[
(\sqrt{1 + d^2})^2 + (\sqrt{e^2 + 1})^2 = (1 + d^2) + (e^2 + 1)
\][/tex]
3. Combine the terms in the parentheses:
[tex]\[
1 + d^2 + e^2 + 1 = 2 + d^2 + e^2
\][/tex]
4. According to the Pythagorean theorem and the given equation:
[tex]\[
2 + d^2 + e^2 = d^2 - 2de + e^2
\][/tex]
5. Compare both sides of the equation:
[tex]\[
2 + d^2 + e^2 \quad \text{and} \quad d^2 - 2de + e^2
\][/tex]
6. Notice that both sides contain [tex]\(d^2 + e^2\)[/tex], so the remaining terms on both sides of the equation are:
[tex]\[
2 \quad \text{and} \quad -2de
\][/tex]
7. Therefore, equate the remaining terms:
[tex]\[
2 = -2de
\][/tex]
Since [tex]\(2\)[/tex] on the left side corresponds to [tex]\(-2de\)[/tex] on the right side of the equation, the missing expression from step 7 is:
[tex]\[
\boxed{-2de}
\][/tex]
Hence, the correct choice is:
B. [tex]\(-2 de\)[/tex]
