To solve the given quadratic equation, we start with:
\[ 4(x^2 + 5) - 17 = 12. \]
First, let's simplify the left side of the equation:
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Distribute the 4: \[ 4x^2 + 20 - 17 = 12. \]
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Combine the constant terms (20 and -17): \[ 4x^2 + 3 = 12. \]
Next, we need to isolate the \(x^2\) term:
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Subtract 3 from both sides: \[ 4x^2 = 12 - 3, \] which simplifies to: \[ 4x^2 = 9. \]
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Now, divide both sides by 4: \[ x^2 = \frac{9}{4}. \]
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To find \(x\), take the square root of both sides: \[ x = \pm \sqrt{\frac{9}{4}}. \]
The square root of a fraction can be simplified: \[ x = \pm \frac{\sqrt{9}}{\sqrt{4}} = \pm \frac{3}{2}. \]
Thus, the solutions for \(x\) are: \[ x = \frac{3}{2} \quad \text{and} \quad x = -\frac{3}{2}. \]
In conclusion, the values of \(x\) in simplest form are: \[ \boxed{\frac{3}{2} \text{ and } -\frac{3}{2}}. \]