1. State the problem: Simplify and expand the expression $$(5m - 1)(x^{2}) - (5m + 2)x + 3m$$. 2. Expand the first term: $$(5m - 1)(x^{2}) = 5m x^{2} - x^{2} = (5m - 1)x^{2}$$. 3. Expand the second term: $$-(5m + 2)x = -5m x - 2x$$. 4. The expression becomes: $$ (5m - 1)x^{2} - 5m x - 2x + 3m$$. 5. This expression is fully expanded. No like terms to combine further. Final answer: $$ (5m - 1)x^{2} - 5m x - 2x + 3m $$.