1. The problem is to expand and simplify the expression $(2x^2 - 5x + 4)(x^2 + 3x - 7)$. 2. Use the distributive property (FOIL for polynomials) to multiply each term in the first polynomial by each term in the second polynomial: $$ (2x^2)(x^2) + (2x^2)(3x) + (2x^2)(-7) + (-5x)(x^2) + (-5x)(3x) + (-5x)(-7) + 4(x^2) + 4(3x) + 4(-7) $$ 3. Calculate each term: $$ 2x^4 + 6x^3 - 14x^2 - 5x^3 - 15x^2 + 35x + 4x^2 + 12x - 28 $$ 4. Combine like terms: $$ 2x^4 + (6x^3 - 5x^3) + (-14x^2 - 15x^2 + 4x^2) + (35x + 12x) - 28 $$ $$ 2x^4 + x^3 - 25x^2 + 47x - 28 $$ 5. The fully expanded and simplified form is: $$ \boxed{2x^4 + x^3 - 25x^2 + 47x - 28} $$