Given two polynomials:
  $$P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$
  $$Q(x) = b_mx^m + b_{m-1}x^{m-1} + ... + b_1x + b_0$$

General Form:

Adding terms with matching degrees:
  $$(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0)$$
  $$ + (b_nx^n + b_{n-1}x^{n-1} + ... + b_1x + b_0)$$
  $$= (a_n + b_n)x^n + (a_{n-1} + b_{n-1})x^{n-1} + ... + (a_0 + b_0)$$

Case 1: Equal Degrees (n = m)

Higher degree terms remain unchanged:
  $$(a_nx^n + ... + a_mx^m + ... + a_1x + a_0)$$
  $$ + (b_mx^m + ... + b_1x + b_0)$$
  $$= a_nx^n + ... + (a_m + b_m)x^m + ... + (a_0 + b_0)$$

Case 2: Different Degrees (n > m)

$$(2x^3 + 3x^2 + x + 1) + (4x^2 + 2x + 3)$$
  $$= 2x^3 + 7x^2 + 3x + 4$$

Example:

To add polynomials, we combine like terms by adding coefficients of terms with the same degree. Let's look at different cases:

Subtracting corresponding terms:
  $$(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0)$$
  $$ - (b_nx^n + b_{n-1}x^{n-1} + ... + b_1x + b_0)$$
  $$= (a_n - b_n)x^n + (a_{n-1} - b_{n-1})x^{n-1} + ... + (a_0 - b_0)$$

Higher degree terms remain unchanged:
  $$(a_nx^n + ... + a_mx^m + ... + a_1x + a_0)$$
  $$ - (b_mx^m + ... + b_1x + b_0)$$
  $$= a_nx^n + ... + (a_m - b_m)x^m + ... + (a_0 - b_0)$$

$$(2x^3 + 3x^2 + x + 1) - (4x^2 + 2x + 3)$$
  $$= 2x^3 - x^2 - x - 2$$

To subtract polynomials, we distribute the minus sign to all terms of the second polynomial and then combine like terms.

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