We are given the equation $10b = 5(\sqrt{c} + 2)$ and need to solve for $c$.

Step-by-step Solution:

1. Start with the original equation: $10b = 5(\sqrt{c} + 2)$
2. Divide both sides by 5 to simplify: $2b = \sqrt{c} + 2$
3. Subtract 2 from both sides: $2b - 2 = \sqrt{c}$
4. Square both sides to eliminate the square root: $(2b - 2)^2 = c$

Thus, the equation we have for $c$ is: $c = (2b - 2)^2$

This matches the given equation, so any equivalent equation must represent the same steps.

Equivalent equation:

Another equivalent form could be: $c = 4(b - 1)^2$

This equation comes from factoring and expanding the expression $(2b - 2)^2$.

Let me know if you would like to dive deeper into any part of this solution!

Relative questions:

1. How do you simplify expressions involving square roots?
2. What are the steps to solve equations involving radicals?
3. How can you expand $(2b - 2)^2$?
4. What techniques are useful in solving quadratic equations?
5. How does squaring both sides of an equation affect its solution?

Tip: Always check for extraneous solutions when squaring both sides of an equation, as it can introduce non-valid answers.