To multiply two polynomials, take the individual terms of one polynomial and multiply each term by the whole of the other polynomial using the distributive quality.
(anxn+an-1xn-1+...+a2x2+a1x+a0)(bmxm+bm-1xm-1+...+b2x2+b1x+b0)=
anxn(bmxm+bm-1xm-1+...+b2x2+b1x+b0)+an-1xn-1(bnxn+bn-1xn-1+...+b2x2+b1x+b0)+...=
anbmxm+n+anbm-1xm+n-1+...+anb2xn+2+anb1xn+1+anb0xn+...
EXAMPLE
(2x2-3x+1)(x3-4x2+5x-6)=
2x2(x3-4x2+5x-6)-3x(x3-4x2+5x-6)+(x3-4x2+5x-6)=
2x5 -8x4 +10x3-12x2
-3x4 +12x3-15x2+18x
x3 -4x2 +5x-6=
2x5-11x4+23x3-31x2 +23x-6.
Note that vertically aligning like powers of x helps when adding the terms. Make sure you have the correct signs in front of each term.