The product of two positive numbers is p. If each of the numbers is increased by 2, by how much (in exact value) is the new product greater than twice the sum of the two original numbers?

Introduction:
This algebraic word problem connects products and sums of two positive numbers expressed in terms of a parameter p. You are told that the product of two numbers is p, and then asked to compare the product after increasing both numbers by 2 with twice the sum of the original numbers. The question tests the ability to write symbolic expressions correctly and then simplify them carefully.

Given Data / Assumptions:

• Let the original numbers be a and b, both positive.
• Their product is ab = p.
• Each number is increased by 2, giving new numbers (a + 2) and (b + 2).
• New product = (a + 2)(b + 2).
• Twice the sum of the original numbers is 2(a + b).
• We must find (new product) - (twice the original sum).

Concept / Approach:
First, expand the new product using distributive multiplication. Then subtract 2(a + b). Use the relation ab = p to express the answer in terms of p and constants only. This is a straightforward symbolic manipulation question, but it is easy to make sign or expansion mistakes if you rush.

Step-by-Step Solution:
Given: ab = p.New product = (a + 2)(b + 2).Expand: (a + 2)(b + 2) = ab + 2a + 2b + 4.Replace ab with p: new product = p + 2a + 2b + 4.Twice the sum of original numbers = 2(a + b) = 2a + 2b.Difference required = (new product) - (twice original sum).Difference = (p + 2a + 2b + 4) - (2a + 2b).Simplify: p + 2a + 2b + 4 - 2a - 2b = p + 4.So the new product is (p + 4) greater than twice the sum of the original numbers.

Verification / Alternative check:
Take a simple numeric example: let a = 2 and b = 3. Then p = ab = 6. New product = (2 + 2)(3 + 2) = 4 * 5 = 20. Twice the sum of originals = 2(2 + 3) = 2 * 5 = 10. Difference = 20 - 10 = 10. Our formula gives p + 4 = 6 + 4 = 10, which matches the calculation.

Why Other Options Are Wrong:
p or 2p: these ignore the constant increase that comes from adding 2 to both numbers.p + 2: underestimates the true difference, missing part of the constant term from the expansion.2p + 4: doubles p unnecessarily, not matching the derived expression.

Common Pitfalls:
Incorrectly expanding (a + 2)(b + 2), for example forgetting the +4 term.Not substituting ab = p correctly, or substituting at the wrong stage.Misinterpreting “how much greater” as a multiplicative comparison instead of a difference.

Final Answer:
p + 4