### Instruction: Solve the inequality
\[-4x^2 + 7x + 2 < 0.\].### Response: To solve a quadratic inequality, I usually start by finding the zeros of the corresponding quadratic equation.
That is, I solve \[-4x^2 + 7x + 2 = 0.\]
I can use the quadratic formula to find the zeros.
The formula says that the zeros are \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\] where $a = -4$, $b = 7$, and $c = 2$.
Plugging these values into the formula, I get \[x = \frac{-7 \pm \sqrt{49 + 32}}{-8}.\]
Simplifying the expression, I get \[x = \frac{-7 \pm \sqrt{81}}{-8}.\]
The square root of $81$ is $9$, so I can write \[x = \frac{-7 \pm 9}{-8}.\]
This gives me two possible values for $x$.
If I choose the plus sign, I get \[x = \frac{-7 + 9}{-8} = \frac{2}{-8} = -\frac{1}{4}.\]
If I choose the minus sign, I get \[x = \frac{-7 - 9}{-8} = \frac{-16}{-8} = 2.\]
So the zeros of the quadratic equation are $x = -\frac{1}{4}$ and $x = 2$.
These are also the points where the quadratic function changes sign.
To see this, I can graph the function using a calculator or some other tool.
I see that the graph is a downward-opening parabola that crosses the x-axis at $x = -\frac{1}{4}$ and $x = 2$.
This means that the function is positive between the zeros and negative outside of them.
Therefore, the inequality is satisfied when $x < -\frac{1}{4}$ or $x > 2$.
These are the solutions to the inequality.