\( (3x^{4}-x^{5}+6x)\cdot .(-3x+9x^{6}-6x^{4}) \)

To expand and simplify the expression \( (3x^{4}-x^{5}+6x)\cdot (-3x+9x^{6}-6x^{4}) \), we can use the distributive property (also known as the FOIL method for binomials). Here’s how it breaks down: 1. Distribute each term in the first polynomial by each term in the second polynomial: - \( 3x^4 \cdot (-3x) = -9x^5 \) - \( 3x^4 \cdot 9x^6 = 27x^{10} \) - \( 3x^4 \cdot (-6x^4) = -18x^8 \) - \( -x^5 \cdot (-3x) = 3x^6 \) - \( -x^5 \cdot 9x^6 = -9x^{11} \) - \( -x^5 \cdot (-6x^4) = 6x^9 \) - \( 6x \cdot (-3x) = -18x^2 \) - \( 6x \cdot 9x^6 = 54x^{7} \) - \( 6x \cdot (-6x^4) = -36x^{5} \) 2. Now combine like terms: - From the degree 11: \(-9x^{11}\) - From the degree 10: \(27x^{10}\) - From the degree 9: \(6x^{9}\) - From the degree 8: \(-18x^{8}\) - From the degree 7: \(54x^{7}\) - From the degree 6: \(3x^{6}\) - From the degree 5: \(-9x^{5} - 36x^{5} = -45x^{5}\) - From the degree 2: \(-18x^{2}\) Putting it all together, the simplified form of the given expression is: \[ -9x^{11} + 27x^{10} + 6x^{9} - 18x^{8} + 54x^{7} + 3x^{6} - 45x^{5} - 18x^{2} \]