1. **State the problem:** We need to find the product of the binomials $ (7x - 3)(5x + 2) $. 2. **Formula used:** To multiply two binomials, use the distributive property (FOIL method): $$ (a + b)(c + d) = ac + ad + bc + bd $$ 3. **Apply the formula:** $$ (7x - 3)(5x + 2) = 7x \cdot 5x + 7x \cdot 2 - 3 \cdot 5x - 3 \cdot 2 $$ 4. **Calculate each term:** $$ 7x \cdot 5x = 35x^2 $$ $$ 7x \cdot 2 = 14x $$ $$ -3 \cdot 5x = -15x $$ $$ -3 \cdot 2 = -6 $$ 5. **Combine like terms:** $$ 35x^2 + 14x - 15x - 6 = 35x^2 + \cancel{14x - 15x} - 6 = 35x^2 - x - 6 $$ 6. **Final answer:** $$ \boxed{35x^2 - x - 6} $$