1. **State the problem:** Solve the equation $$ (2x - 1)(x - 1) + (4x + 3)(x - 4) = (3x - 5)(2x - 5) $$. 2. **Expand each product:** $$ (2x - 1)(x - 1) = 2x^2 - 2x - x + 1 = 2x^2 - 3x + 1 $$ $$ (4x + 3)(x - 4) = 4x^2 - 16x + 3x - 12 = 4x^2 - 13x - 12 $$ $$ (3x - 5)(2x - 5) = 6x^2 - 15x - 10x + 25 = 6x^2 - 25x + 25 $$ 3. **Rewrite the equation with expanded terms:** $$ (2x^2 - 3x + 1) + (4x^2 - 13x - 12) = 6x^2 - 25x + 25 $$ 4. **Combine like terms on the left side:** $$ 2x^2 + 4x^2 - 3x - 13x + 1 - 12 = 6x^2 - 25x + 25 $$ $$ 6x^2 - 16x - 11 = 6x^2 - 25x + 25 $$ 5. **Subtract $6x^2$ from both sides:** $$ \cancel{6x^2} - 16x - 11 = \cancel{6x^2} - 25x + 25 $$ $$ -16x - 11 = -25x + 25 $$ 6. **Add $25x$ to both sides:** $$ -16x + 25x - 11 = -25x + 25 + 25x $$ $$ 9x - 11 = 25 $$ 7. **Add 11 to both sides:** $$ 9x - 11 + 11 = 25 + 11 $$ $$ 9x = 36 $$ 8. **Divide both sides by 9:** $$ x = \frac{36}{9} $$ $$ x = 4 $$ **Final answer:** $$ x = 4 $$