To expand the expression \((x + 2)(3x^2 + 12) + (x - 2)\), we first work on the product \((x + 2)(3x^2 + 12)\).
Using the distributive property (also known as the FOIL method for binomials), we will distribute \(x\) and \(2\) across the terms in \(3x^2 + 12\):
\[ (x + 2)(3x^2 + 12) = x(3x^2 + 12) + 2(3x^2 + 12) \]
Now, calculate each part:
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For \(x(3x^2 + 12)\): \[ x \cdot 3x^2 = 3x^3 \] \[ x \cdot 12 = 12x \] Thus, \[ x(3x^2 + 12) = 3x^3 + 12x \]
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For \(2(3x^2 + 12)\): \[ 2 \cdot 3x^2 = 6x^2 \] \[ 2 \cdot 12 = 24 \] Therefore, \[ 2(3x^2 + 12) = 6x^2 + 24 \]
Putting it all together: \[ (x + 2)(3x^2 + 12) = 3x^3 + 12x + 6x^2 + 24 \]
Now we combine everything: \[ 3x^3 + 6x^2 + 12x + 24 + (x - 2) \]
We expand \((x - 2)\): \[ x - 2 \]
Now combine all the terms: \[ 3x^3 + 6x^2 + 12x + 24 + x - 2 \]
Combine the like terms:
- Combine \(12x\) and \(x\): \[ 12x + x = 13x \]
- Combine \(24\) and \(-2\): \[ 24 - 2 = 22 \]
Thus, the final result is: \[ 3x^3 + 6x^2 + 13x + 22 \]
So, the expanded form of the expression is: \[ \boxed{3x^3 + 6x^2 + 13x + 22} \]