### Instruction: Solve the inequality
\[-4x^2 + 7x + 2 < 0.\].### Response: To solve a quadratic inequality, I need to find the values of x that make the quadratic expression negative.
To do that, I first need to find the zeros of the quadratic expression, since those are the points where it changes sign.
I can use the quadratic formula to find the zeros of \[-4x^2 + 7x + 2.\]
The quadratic formula says that the zeros are given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\] where a, b, and c are the coefficients of the quadratic expression.
In this case, a = -4, b = 7, and c = 2, so plugging those into the formula, I get \[x = \frac{-7 \pm \sqrt{7^2 - 4(-4)(2)}}{2(-4)},\] which simplifies to \[x = \frac{-7 \pm \sqrt{81}}{-8}.\]
The square root of 81 is 9, so I have \[x = \frac{-7 \pm 9}{-8},\] which gives me two possible values for x.
One value is \[x = \frac{-7 + 9}{-8} = \frac{2}{-8} = -\frac{1}{4},\] and the other value is \[x = \frac{-7 - 9}{-8} = \frac{-16}{-8} = 2.\]
These are the two zeros of the quadratic expression, and they divide the number line into three intervals: x < -1/4, -1/4 < x < 2, and x > 2.
To check the sign of the quadratic expression in each interval, I can pick any value in that interval and plug it into the expression.
For example, if I pick x = -1, which is in the interval x < -1/4, I get \[-4(-1)^2 + 7(-1) + 2 = -4 - 7 + 2 = -9,\] which is negative.
This means that the quadratic expression is negative for all values of x in the interval x < -1/4.
Similarly, I can pick x = 0, which is in the interval -1/4 < x < 2, and get \[-4(0)^2 + 7(0) + 2 = 2,\] which is positive.
This means that the quadratic expression is positive for all values of x in the interval -1/4 < x < 2.
Finally, I can pick x = 3, which is in the interval x > 2, and get \[-4(3)^2 + 7(3) + 2 = -36 + 21 + 2 = -13,\] which is negative.
This means that the quadratic expression is negative for all values of x in the interval x > 2.
Therefore, the solution to the inequality is the union of the intervals where the expression is negative, which is \[x < -\frac{1}{4} \text{ or } x > 2.\]