To find the solutions to the quadratic equation \( 4x^2 + 2x = 10 \), we first rearrange it into standard form:
\[ 4x^2 + 2x - 10 = 0 \]
Now we can apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = 2 \), and \( c = -10 \).
Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4(4)(-10) = 4 + 160 = 164 \]
Now we can substitute the values into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{164}}{2(4)} = \frac{-2 \pm \sqrt{164}}{8} \]
Since \( \sqrt{164} = \sqrt{4 \times 41} = 2\sqrt{41} \):
\[ x = \frac{-2 \pm 2\sqrt{41}}{8} = \frac{-2}{8} \pm \frac{2\sqrt{41}}{8} = \frac{-1 \pm \sqrt{41}}{4} \]
We can separate this into the two solutions:
\[ x_1 = \frac{-1 + \sqrt{41}}{4} \quad \text{and} \quad x_2 = \frac{-1 - \sqrt{41}}{4} \]
From the provided options, the correct solutions must match these two forms.
Thus, the solution \( \frac{-1 - \sqrt{41}}{4} \) matches one of the options:
- Response: \( -1 - \frac{\sqrt{41}}{4} \)
Therefore, this response is a solution to the quadratic equation.
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