To add two polynomials, you just put them together. For example,
if you had to add −2*x*^{2} +
4*y*^{2} − 2*y* and
−*x**y*^{2} + 3*x*^{2}
− *x*, the first step is just to write them as one
polynomial, getting −2*x*^{2} +
4*y*^{2} − 2*y*
− *x**y*^{2} + 3*x*^{2}
− *x*. (In this case, the second polynomial started
with a minus sign; if it hadn't, then we would have put a plus
sign in between.) Now we have to collect like terms. The only two
terms we can combine are the −2*x*^{2} and the
+ 3*x*^{2}. They combine to *x*^{2},
since −2 + 3 = 1 and 1*x*^{2} is the same thing
as *x*^{2}. So the answer is *x*^{2} +
4*y*^{2} − 2*y*
− *x**y*^{2} − *x*.

If we had to subtract the polynomials instead of adding them,
then we would do almost the same thing, except we would switch the
signs of all the terms of the second polynomial. (Be sure to
switch *all* of them.) Then we would get
−2*x*^{2} + 4*y*^{2} −
2*y* + *x**y*^{2} −
3*x*^{2} + *x*. We still need to collect like
terms, though, so we would end up with
−5*x*^{2} + 4*y*^{2} −
2*y* + *x**y*^{2} + *x*.

Note that these are all still polynomials, even though they have more than one letter in them.