Let's solve the equation step by step:

Given equation:
\[
2(2^x + 2^x + 2^x + 2^x) = 2^{20}
\]

### Step 1: Simplify the expression inside the parentheses
Notice that \(2^x\) is repeated four times inside the parentheses. So, we can factor it out:
\[
2(4 \cdot 2^x) = 2^{20}
\]

### Step 2: Simplify further
Since \(4\) is equal to \(2^2\), we can write:
\[
2(2^2 \cdot 2^x) = 2^{20}
\]

Combine the exponents of \(2\):
\[
2(2^{x+2}) = 2^{20}
\]

### Step 3: Distribute the 2 on the outside
Now, distribute the 2:
\[
2^1 \cdot 2^{x+2} = 2^{20}
\]

Combine the exponents on the left-hand side:
\[
2^{x+3} = 2^{20}
\]

### Step 4: Equate the exponents
Since the bases are the same, you can equate the exponents:
\[
x + 3 = 20
\]

### Step 5: Solve for \(x\)
Subtract 3 from both sides:
\[
x = 17
\]

So, the solution is \(x = 17\).

Would you like more details on any part of this solution? 

Here are some related questions you might find interesting:
1. What is the general solution method for exponential equations?
2. How can logarithms be used to solve exponential equations?
3. What are the properties of exponents that are crucial for solving such equations?
4. How can we solve an equation where the bases are not the same?
5. What happens when there are multiple different bases in the equation?

**Tip:** Always try to factor and simplify expressions inside parentheses before distributing or expanding. This often makes solving the equation much easier.

Understand how to solve the exponential equation 2(2^x + 2^x + 2^x + 2^x) = 2^20 through detailed step-by-step simplification. This problem demonstrates the application of exponentiation rules and algebraic concepts, ideal for students in Grades 9-12.

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