It is straightforward to add or subtract two polynomials. We simply add or subtract terms of the same degree. For example:
Consider two polynomials of degrees m and n. Suppose that we add or subtract the two polynomials. Two cases now arise:

If m and n are unequal, the resulting polynomial will be of a degree equal to the larger of m and n.

If m and n are equal, the resulting polynomial will have a degree which is at the most equal to m and n, but it could also have a degree which is less than m and n.
Let’s understand these facts through the following examples:
(a) Suppose that
\[\begin{array}{l}p\left( x \right):{x^2} + x  1\\q\left( x \right):{x^4}  {x^3}  2\\ \Rightarrow \left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right)} \right) = 2\\{\rm{Deg}}\left( {q\left( x \right)} \right) = 4\end{array} \right.\end{array}\]
Now,
\[\begin{array}{l}p\left( x \right) + q\left( x \right):{x^4}  {x^3} + {x^2} + x  3\\p\left( x \right)  q\left( x \right):  {x^4} + {x^3} + {x^2} + x + 1\\ \Rightarrow \left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right) + q\left( x \right)} \right) = 4\\{\rm{Deg}}\left( {p\left( x \right)  q\left( x \right)} \right) = 4\end{array} \right.\end{array}\]
(b) Suppose that
\[\begin{array}{l}p\left( x \right):{x^4}  {x^2} + 1\\q\left( x \right):{x^4} + {x^2}  1\\ \Rightarrow \left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right)} \right) = 4\\{\rm{Deg}}\left( {q\left( x \right)} \right) = 4\end{array} \right.\end{array}\]
Now,
\[\begin{array}{l}p\left( x \right) + q\left( x \right):2{x^4}\\p\left( x \right)  q\left( x \right):  2{x^2} + 2\\ \Rightarrow \left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right) + q\left( x \right)} \right) = 4\\{\rm{Deg}}\left( {p\left( x \right)  q\left( x \right)} \right) = 2\end{array} \right.\end{array}\]
\[\begin{array}{l}p\left( x \right) + q\left( x \right):2{x^4}\\p\left( x \right)  q\left( x \right):  2{x^2} + 2\\ \Rightarrow \;\;\;\;\;\;\;\left\{ \begin{array}{l}{\rm{Deg}}\left( {p\left( x \right) + q\left( x \right)} \right) = 4\\{\rm{Deg}}\left( {p\left( x \right)  q\left( x \right)} \right) = 2\end{array} \right.\end{array}\]
Thus, in the case that the two polynomials have equal degrees, we see that there is a possibility of the highest degree terms cancelling each other out upon addition or subtraction, and so the resulting polynomial might have a degree less than the original degrees of the two polynomials.