There is a "cubic formula" (in fact two versions, a radical version and one using trig functions) and a "quartic formula" but they are usually more trouble than less powerful methods.

If there are repeating patterns in a polynomial they may indicate a factorization in the same way that (say) 424242 = 10101 * 42.
This method will work on the first problem - you can see a factor of x-4.

Once you find one factor, divide it out and the quotient will be lower-order.

IF the coefficients of the factors are as (Ax+a)(Bx+b)(Cx+c)
then the cubic term is ABCx³ and the constant term is abc. If we suppose that A,B,C,a,b,c are integers, this gets us down to a small number of possibilities; we then check the quadratic term

[ABc+AbC+aBC]x²

and the linear term

[abC+aBc+Abc]x

to see which combination works. That should get the second one (not without some effort!)

Finally, a quartic with no odd (x or x³) terms can often be solved by setting y=x² to get a quadratic in y. The roots of this may be of the form Ax² + a and irreducible; or they may be of the form Ax² - a and factor further. This will work for your example.

Over the reals not every quadratic factors; you can use other tricks such as

x⁴ + x² + 1 = (x⁴ + 2x² +1) - x²
= [(x² + 1) + x][x² +1 -x]
= [x²+x+1][x²-x+1]

Good Hunting!
RD