\( (3x^{4}-x^{5}+6x^{2})\cdot .(-3x+9x^{6}-6x^{4}) \)

To understand the multiplication of these two polynomial expressions, let's first organize them nicely. We have \( (3x^{4}-x^{5}+6x^{2}) \) and \( (-3x+9x^{6}-6x^{4}) \). When multiplying polynomials, apply the distributive property (often referred to as the FOIL method for binomials, but it applies here as well). First, distribute each term in the first polynomial to each term in the second polynomial. This leads to a series of multiplications: - Multiply \( 3x^{4} \) by each term in the second polynomial: - \( 3x^{4} \cdot (-3x) = -9x^{5} \) - \( 3x^{4} \cdot 9x^{6} = 27x^{10} \) - \( 3x^{4} \cdot (-6x^{4}) = -18x^{8} \) - Next, multiply \( -x^{5} \) by each term in the second polynomial: - \( -x^{5} \cdot (-3x) = 3x^{6} \) - \( -x^{5} \cdot 9x^{6} = -9x^{11} \) - \( -x^{5} \cdot (-6x^{4}) = 6x^{9} \) - Finally, multiply \( 6x^{2} \) by each term in the second polynomial: - \( 6x^{2} \cdot (-3x) = -18x^{3} \) - \( 6x^{2} \cdot 9x^{6} = 54x^{8} \) - \( 6x^{2} \cdot (-6x^{4}) = -36x^{6} \) Now, combine all those products: \[ -9x^{5} + 27x^{10} - 18x^{8} + 3x^{6} - 9x^{11} + 6x^{9} - 18x^{3} + 54x^{8} - 36x^{6} \] Grouping like terms together: \[ -9x^{11} + 27x^{10} + (6x^{9}) + (-9x^{5} + 3x^{6} - 36x^{6}) + (-18x^{8} + 54x^{8}) - 18x^{3} \] \[ = -9x^{11} + 27x^{10} + 6x^{9} - 9x^{5} - 33x^{6} + 36x^{8} - 18x^{3} \] Thus, when fully simplified, the expression becomes: \[ -9x^{11} + 27x^{10} + 6x^{9} + 36x^{8} - 33x^{6} - 9x^{5} - 18x^{3} \]