Solving polynomials greater than quadratics is actually quite difficult and not something one will likely have to do aside from special cases where the polynomial can be reduced to a quadratic.

A quadratic is an equation of the form ax^{2}+bx+c=0, and as you may or may not know can be solved by the following formula: x=(-b±√(b^{2}-4ac))/2a.

Adding and subtracting polynomials is quite simple, as it just involves combining like terms. For example, say I have two quadratic polynomial functions, p_{1}(x)=2x^{2}+3x+1 and p_{2}(x)=x^{2}+4x-2, then p_{1}(x)+p_{2}(x)=(2+1)x^{2}+(3+4)x+(1-2)=3x^{2}+7x-1, the same goes for subtracting polynomials.

Multiplying polynomials is a bit more difficult as it involves using FOIL (Front, Outside, Inside, Last) multiplication. For example lets take two simple polynomials, a(x)=x^{2}+1 and b(x)=x^{3}+2.

a(x)*b(x)=(x^{2}+1)(x^{3}+2)=(x^{2}*x^{3})+(x^{2}*2)+(1*x^{3})+(1*2)=x^{5}+x^{3}+2x^{2}+2