Two numbers have sum 8 and difference 4. Form the quadratic equation having these numbers as roots, and pick the correct option.

Introduction / Context: From the sum and difference of two numbers, you can determine the numbers and then the quadratic having them as roots. This illustrates reverse construction of a polynomial from root properties.

Given Data / Assumptions:

• Sum S = 8.
• Difference D = 4 (so numbers are (S ± D)/2).
• Quadratic desired with those numbers as roots.

Concept / Approach: Compute the two numbers, then use x^2 − Sx + P = 0, where P is their product. Alternatively, directly use S and P without computing the numbers individually.

Step-by-Step Solution:

Verification / Alternative check: Factor: (x − 6)(x − 2) = x^2 − 8x + 12 ⇒ roots 6 and 2, consistent.

Why Other Options Are Wrong: Sign errors in the x-term or constant term change the sum/product, creating different roots.

Common Pitfalls: Taking D itself as a root or misusing S and P in the formula.

Final Answer: x^2 − 8x + 12 = 0