1. **State the problem:** Multiply the binomials $ (x + 19)(3x + 1) $. 2. **Formula used:** To multiply two binomials, use the distributive property (also known as FOIL method for binomials): $$ (a + b)(c + d) = ac + ad + bc + bd $$ 3. **Apply the formula:** $$ (x + 19)(3x + 1) = x \cdot 3x + x \cdot 1 + 19 \cdot 3x + 19 \cdot 1 $$ 4. **Calculate each term:** $$ = 3x^2 + x + 57x + 19 $$ 5. **Combine like terms:** $$ x + 57x = 58x $$ 6. **Final expression:** $$ 3x^2 + 58x + 19 $$ This is the product of the two binomials.