To simplify the given expression, we will combine like terms. The expression is:

$25798901760 x^{15} + 693345484800 x^{14} + 8115994091520 x^{13} + 53215051645440 x^{12} + 209637356788224 x^{11} + 492901990565184 x^{10} + 633914720020224 x^{9} + 330212258682354 x^{8} - 25078464308256 x^{7}$ $-(537477120 x^{13} + 10592778240 x^{12} + 86266944000 x^{11} + 344449304064 x^{10} + 648770329440 x^{9} + 430886211768 x^{8} - 40689774432 x^{7})$

Step-by-step Simplification:

1. Distribute the negative sign to each term inside the parentheses:

$25798901760 x^{15} + 693345484800 x^{14} + 8115994091520 x^{13} + 53215051645440 x^{12} + 209637356788224 x^{11} + 492901990565184 x^{10} + 633914720020224 x^{9} + 330212258682354 x^{8} - 25078464308256 x^{7}$ $- 537477120 x^{13} - 10592778240 x^{12} - 86266944000 x^{11} - 344449304064 x^{10} - 648770329440 x^{9} - 430886211768 x^{8} + 40689774432 x^{7}$

2. Now combine like terms for each power of $x$.

• For $x^{15}$:
  $25798901760 x^{15}$
  (No other terms for $x^{15}$, so this stays the same.)

• For $x^{14}$:
  $693345484800 x^{14}$
  (No other terms for $x^{14}$, so this stays the same.)

• For $x^{13}$:
  $8115994091520 x^{13} - 537477120 x^{13} = 8115994091520 x^{13} - 537477120 x^{13} = 7578516974400 x^{13}$

• For $x^{12}$:
  $53215051645440 x^{12} - 10592778240 x^{12} = 53215051645440 x^{12} - 10592778240 x^{12} = 53204458867200 x^{12}$

• For $x^{11}$:
  $209637356788224 x^{11} - 86266944000 x^{11} = 209637356788224 x^{11} - 86266944000 x^{11} = 209551089844224 x^{11}$

• For $x^{10}$:
  $492901990565184 x^{10} - 344449304064 x^{10} = 492901990565184 x^{10} - 344449304064 x^{10} = 492557541261120 x^{10}$

• For $x^{9}$:
  $633914720020224 x^{9} - 648770329440 x^{9} = 633914720020224 x^{9} - 648770329440 x^{9} = 633914071249792 x^{9}$

• For $x^{8}$:
  $330212258682354 x^{8} - 430886211768 x^{8} = 330212258682354 x^{8} - 430886211768 x^{8} = 329781372471586 x^{8}$

• For $x^{7}$:
  $-25078464308256 x^{7} + 40689774432 x^{7} = -25078464308256 x^{7} + 40689774432 x^{7} = -25037774533824 x^{7}$

Final Simplified Expression:

$25798901760 x^{15} + 693345484800 x^{14} + 7578516974400 x^{13} + 53204458867200 x^{12} + 209551089844224 x^{11} + 492557541261120 x^{10} + 633914071249792 x^{9} + 329781372471586 x^{8} - 25037774533824 x^{7}$

Would you like more details on the steps or have any questions about this simplification?

Here are some related questions:

1. How do you factor polynomials with multiple terms?
2. What is the general method for combining like terms in polynomials?
3. How can you simplify an expression with exponents greater than 1?
4. What are the rules for distributing negative signs across terms in an expression?
5. How do you solve polynomial equations?

Tip: When simplifying polynomials, always make sure to check the exponents of the terms to ensure that you're combining the correct ones!