You have to multiply every term by every other term. A good way to make sure you don't miss any is to use a table.
First, combine like terms within each group of parentheses: $(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)\\ =(3x^4-7x^3+5x^2+5x+8)(-4x^2+3x+9)$
Then form a table and multiply each term by multiplying the coefficients and adding the exponents: $ \begin{array}{c|cc} \text{} & 3x^4 & -7x^3 & 5x^2 & 5x & 8 \\ \hline -4x^2 & -12x^6 & -28x^5 & -20x^4 & -20x^3 & -32x^2 \\ 3x & 9x^5 & -21x^4 & 15x^3 & 15x^2 & 24x \\ 9 & 27x^4 & -63x^3 & 45x^2 & 45x & 72 \\ \end{array} $
Now take the new polynomial from the table and combine like terms: $-12x^6-28x^5-20x^4-20x^3-32x^2+9x^5-21x^4+15x^3\\ +15x^2+24x+27x^4-63x^3+45x^2+45x+72\\$$ =-12x^6+(9-28)x^5+(27-21-20)x^4+(-63+15-20)x^3\\+(45+15-32)x^2+(45+24)x+72\\$$ =-12x^6-19x^5-14x^4-68x^3+28x^2+69x+72$
This method will also work with negative and non-integer exponents, as it is not restricted to polynomials.