To find the product of the expressions \((4x^{10}+9x^{7})(-4x^{10}-9x^{7})\), we can use the distributive property (also known as the FOIL method for binomials).
Let's denote:
- \(A = 4x^{10}\)
- \(B = 9x^{7}\)
So, we can rewrite the expression as:
\[(A + B)(-A - B)\]
Now, applying the distributive property:
\[ = A \cdot (-A) + A \cdot (-B) + B \cdot (-A) + B \cdot (-B) \]
Calculating each term:
- \(A \cdot (-A) = -A^2 = - (4x^{10})^2 = -16x^{20}\)
- \(A \cdot (-B) = -AB = - (4x^{10})(9x^{7}) = -36x^{17}\)
- \(B \cdot (-A) = -BA = - (9x^{7})(4x^{10}) = -36x^{17}\) (same as above)
- \(B \cdot (-B) = -B^2 = - (9x^{7})^2 = -81x^{14}\)
Now, combine the terms:
\[ -16x^{20} - 36x^{17} - 36x^{17} - 81x^{14} \]
Combine the like terms \((-36x^{17} - 36x^{17} = -72x^{17})\):
\[ -16x^{20} - 72x^{17} - 81x^{14} \]
Thus, the final answer is:
\[ \boxed{-16x^{20} - 72x^{17} - 81x^{14}} \]