Answer: guess roots and factor.

Quadratic equations are polynomial equations whose highest power is 2, while **quartic equations** are polynomial equations whose highest power is 4.

Any polynomial expression (whether quadratic, quartic, or any other power) can be factored into its roots. A root is a value which, when plugged in for the variable, equals zero.

Example: ##x^4 + 3x^3 + x^2 – x + 2 = 2x + 4##.

First rewrite as a polynomial that equals 0: ##x^4 + 3x^3 + x^2 -3x -2 = 0##.

A quartic equation has at most 4 roots. It isn’t elegant, but we can find them by guessing a value for ##x## and seeing whether it equals zero. 1 works here because ##1^4 + 3*1^3 + 1^2 – 3*1 – 2 = 0##.

Now that we have one root, 1, we know that the expression can be factored into ##(x-1)## and a cubic because plugging 1 into the ##(x-1)## term equals 0.

To find the cubic, we use polynomial long division:

##(x^4+3x^3+x^2-3x-2)/(x-1) = x^3 + 4x^2 + 5x + 2##

Continuing, we can find the other roots -1 (a double root) and -2. Or you can use the cubic or quadratic equation once you’ve reduced the polynomial.

This gives the final factorization of ##(x-1)(x+1)^2(x+2) = 0##. So the solutions are ##x = 1, -1, text{or} -2##.