To solve the equation \(8 = 4(4 + 2n)\), we'll first apply the properties of equality step by step.
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Distribute the 4 on the right side: \[ 8 = 4 \cdot 4 + 4 \cdot 2n \] This simplifies to: \[ 8 = 16 + 8n \]
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Subtract 16 from both sides to isolate the term with \(n\): \[ 8 - 16 = 8n \] This simplifies to: \[ -8 = 8n \]
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Divide both sides by 8 to solve for \(n\): \[ n = \frac{-8}{8} \] This simplifies to: \[ n = -1 \]
Now, we have the solution \(n = -1\).
Next, we check our solution by substituting \(n = -1\) back into the original equation:
Substitution: \[ 8 = 4(4 + 2(-1)) \] Calculate the expression inside the parentheses: \[ 8 = 4(4 - 2) \] This simplifies to: \[ 8 = 4 \cdot 2 \] And further simplifies to: \[ 8 = 8 \]
Since both sides of the equation are equal, our solution is confirmed to be correct.
The final answer is: \[ \boxed{-1} \]