We are given the quadratic equation [tex]\( -2x^2 + x + 4 = 0 \)[/tex]. To find the solutions, we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In our equation [tex]\( -2x^2 + x + 4 = 0 \)[/tex], the coefficients are:
[tex]\[
a = -2, \quad b = 1, \quad c = 4
\][/tex]
First, we calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (1)^2 - 4(-2)(4)
\][/tex]
[tex]\[
\Delta = 1 + 32
\][/tex]
[tex]\[
\Delta = 33
\][/tex]
Now, we substitute back into the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting [tex]\( b = 1 \)[/tex] and [tex]\( \Delta = 33 \)[/tex]:
[tex]\[
x = \frac{-1 \pm \sqrt{33}}{2(-2)}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{33}}{-4}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{33}}{-4} = \frac{1 \mp \sqrt{33}}{4}
\][/tex]
Therefore, the solutions are:
[tex]\[
x = \frac{1 + \sqrt{33}}{4} \quad \text{and} \quad x = \frac{1 - \sqrt{33}}{4}
\][/tex]
Comparing this to the given choices, we see that the correct answer is:
[tex]\[
\boxed{B. \, x = \frac{1 \pm \sqrt{33}}{4}}
\][/tex]
