Solve for x :
`3(2^(x) + 1) - 2^(x + 2)) + 5 = 0`.

Answer

To solve the equation \( 3(2^x + 1) - 2^{x + 2} + 5 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation for clarity:
\[ 3(2^x + 1) - 2^{x + 2} + 5 = 0 \]

### Step 2: Distribute the 3
Distributing the 3 into the parentheses gives:
\[ 3 \cdot 2^x + 3 - 2^{x + 2} + 5 = 0 \]

### Step 3: Combine like terms
Combine the constant terms (3 and 5):
\[ 3 \cdot 2^x - 2^{x + 2} + 8 = 0 \]

### Step 4: Rewrite \(2^{x + 2}\)
Recall that \(2^{x + 2} = 2^x \cdot 2^2 = 4 \cdot 2^x\). Substitute this back into the equation:
\[ 3 \cdot 2^x - 4 \cdot 2^x + 8 = 0 \]

### Step 5: Combine the terms with \(2^x\)
Now combine the terms involving \(2^x\):
\[ (3 - 4) \cdot 2^x + 8 = 0 \]
This simplifies to:
\[ -1 \cdot 2^x + 8 = 0 \]

### Step 6: Isolate \(2^x\)
Rearranging gives:
\[ -2^x = -8 \]
Thus:
\[ 2^x = 8 \]

### Step 7: Rewrite 8 as a power of 2
Recognizing that \(8 = 2^3\), we can equate the exponents:
\[ 2^x = 2^3 \]

### Step 8: Solve for \(x\)
Since the bases are the same, we can set the exponents equal to each other:
\[ x = 3 \]

### Final Answer
Thus, the solution is:
\[ x = 3 \]

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