To solve the equation \( 4(5x + 3) - 2x + 2 = 32 \), we will first simplify the left side of the equation step by step.
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Distribute the \( 4 \) in \( 4(5x + 3) \):
\[ 4 \cdot 5x + 4 \cdot 3 = 20x + 12 \]
So the equation now looks like:
\[ 20x + 12 - 2x + 2 = 32 \]
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Combine like terms on the left side:
\[ (20x - 2x) + (12 + 2) = 18x + 14 \]
Now our equation is:
\[ 18x + 14 = 32 \]
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Next, we isolate \( x \) by subtracting \( 14 \) from both sides:
\[ 18x = 32 - 14 \]
\[ 18x = 18 \]
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Finally, divide both sides by \( 18 \):
\[ x = \frac{18}{18} \]
\[ x = 1 \]
Thus, the solution to the equation is:
\[ \boxed{1} \]