## On the Subject of Increasing Indices

"No one uses these equations in real life ..." - someone who blew up, probably.

The module shows a prototype version of the **Explosive Equation Generator 3000**, a machine developed by
a team of radical mathematics teachers so infuriated by their students' lack of interest in solving
basic polynomials^{*} that they started designing machines for a bomb manufacture company. The result is
this device, designed to torture those incapable of solving equations by endlessly demanding
solutions to equations of continually increasing degree until the defusal team makes a mistake or is
pushed to the point of insanity - whichever comes first.

The device features a display showing a polynomial expression and 8 buttons, numbered from -3 to 4. To disarm the module, solve the polynomial equations created by the expressions which appear on the display. As this is a prototype of the device, only three equations need to be solved in total: one quadratic, one cubic, and one quartic. For each expression, set it equal to 0 and solve the equation to find all solutions for x. Press all buttons corresponding to solutions to progress to the next expression.

Pressing any button which is **not** a valid solution for x will incur a strike, and a
new equation of the same degree will be generated. Equations which have already been solved will not be lost.
It is not necessary to press repeated roots multiple times, but there is no penalty for doing so.

** ^{*}**Relevant mathematical definitions:

**Polynomial:**an equation consisting of variables and coefficients where the only operations used are addition, subtraction, multiplication, and non-negative powers. For example, x^{2}+4x+2=0.**Coefficient:**the number preceding a term in an equation. For example, if a term is 5x, the coefficient is 5.**Term:**A part of an equation between operators, for example in 3x + 7 the terms are 3x and 7.