To solve the quadratic equation \( 4x^2 + 2x = 10 \), we first need to set it equal to zero:
\[ 4x^2 + 2x - 10 = 0. \]
Next, we can apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 4 \), \( b = 2 \), and \( c = -10 \).
Now, we can calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 2^2 - 4(4)(-10) = 4 + 160 = 164. \]
Next, we can substitute the values into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{164}}{2 \times 4} = \frac{-2 \pm \sqrt{164}}{8}. \]
We can simplify \( \sqrt{164} \):
\[ \sqrt{164} = \sqrt{4 \times 41} = 2\sqrt{41}. \]
Substituting this back into the formula gives:
\[ x = \frac{-2 \pm 2\sqrt{41}}{8} = \frac{-1 \pm \sqrt{41}}{4}. \]
Thus, the two solutions to the equation \( 4x^2 + 2x = 10 \) are:
\[ x = \frac{-1 + \sqrt{41}}{4} \quad \text{and} \quad x = \frac{-1 - \sqrt{41}}{4}. \]
These can be considered as the solutions to your quadratic equation. If you have specific options you want to compare against for the solution, please provide those options.
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