1. Stating the problem: Expand and simplify the product $$ (x^3 - 2x^2 + 5x - 7)(2x - 3) $$ 2. Distribute each term in the second polynomial to every term in the first polynomial: $$ (x^3 - 2x^2 + 5x - 7)(2x) - (x^3 - 2x^2 + 5x - 7)(3) $$ 3. Multiply term-by-term: $$ 2x imes x^3 = 2x^4 $$ $$ 2x imes (-2x^2) = -4x^3 $$ $$ 2x imes 5x = 10x^2 $$ $$ 2x imes (-7) = -14x $$ $$ -3 imes x^3 = -3x^3 $$ $$ -3 imes (-2x^2) = 6x^2 $$ $$ -3 imes 5x = -15x $$ $$ -3 imes (-7) = 21 $$ 4. Write all terms together: $$ 2x^4 - 4x^3 + 10x^2 - 14x - 3x^3 + 6x^2 - 15x + 21 $$ 5. Combine like terms: $$ 2x^4 + (-4x^3 - 3x^3) + (10x^2 + 6x^2) + (-14x - 15x) + 21 $$ $$ = 2x^4 - 7x^3 + 16x^2 - 29x + 21 $$ Final answer: $$ \boxed{2x^4 - 7x^3 + 16x^2 - 29x + 21} $$ This is the expanded and simplified form of the product.