1. **Problem statement:** We are given two numbers whose product is $ab$. We want to find the increase in the product when the first number is increased by 4. 2. **Initial product:** The product of the two numbers is given by $$P = ab$$ 3. **New product after increasing the first number by 4:** If the first number is increased by 4, the new first number is $a + 4$. The second number remains $b$. So the new product is $$P_{new} = (a + 4) \times b$$ 4. **Calculate the increase in product:** The increase in the product is the difference between the new product and the original product: $$\text{Increase} = P_{new} - P = (a + 4)b - ab$$ 5. **Simplify the expression:** $$\text{Increase} = ab + 4b - ab = 4b$$ 6. **Answer:** The increase in the product when the first number is increased by 4 is $4b$. Therefore, the correct choice is **(a) 4b**.