Calculate Edge Length AE in a Rectangular Prism Using √(26a²+8a+16)

To solve this problem of calculating $AE$ in the rectangular prism, we will use the Pythagorean Theorem:

Step 1: Identify given variables:
We know $AF = \sqrt{26a^2 + 8a + 16}$ and $HG = a + 4$.

Step 2: Problem Setup:
We recognize that $AF$ is a diagonal across the face $AFE$ in the prism. Given expressions can be linked: $(AE)^2 + (EF)^2 = (AF)^2$.

Step 3: Utilize $HG$:
Since $HG = a + 4$, $EF = HG = a + 4$ as triangles and prisms share dimensions proportionally, so $(AE)^2 + (a + 4)^2 = 26a^2 + 8a + 16$.

Step 4: Equation Simplification:
We'll expand and simplify further:
$(AE)^2 + a^2 + 8a + 16 = 26a^2 + 8a + 16.$
Solving gives us $(AE)^2 = 25a^2$.

Step 5: Solution Conclusion:
Calculate $AE$ as $\sqrt{25a^2} = 5a$, through recognizing $AE$ only needs formula $\sqrt{x}$ operation once simplified.

Therefore, the calculated length $AE$ is $5a$.