To solve the quadratic equation \(4x^2 + 2x = 10\), we first rearrange it into standard form:
\[ 4x^2 + 2x - 10 = 0 \]
Next, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 4\), \(b = 2\), and \(c = -10\).
Now we calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 - 4ac = 2^2 - 4 \cdot 4 \cdot (-10) = 4 + 160 = 164 \]
Now substitute the values into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{164}}{2 \cdot 4} \]
Calculating \(\sqrt{164}\):
\[ \sqrt{164} \approx 12.81 \]
So we find:
\[ x = \frac{-2 \pm 12.81}{8} \]
Calculating the two possible values for \(x\):
- \(x = \frac{-2 + 12.81}{8} = \frac{10.81}{8} \approx 1.35\)
- \(x = \frac{-2 - 12.81}{8} = \frac{-14.81}{8} \approx -1.85\)
Thus, the approximate solutions to the equation rounded to the nearest hundredth are:
\[ x \approx 1.35 \quad \text{and} \quad x \approx -1.85 \]
From the given options, the solution \(x = -1.85\) is the correct one.