Quadratic construction: Find the monic quadratic equation whose real roots are 3 and −1. Choose the correct standard-form equation.

Introduction / Context:
The question asks you to build a quadratic equation from its roots. This is a standard skill in algebra: once the roots are known, the quadratic can be written quickly using the factor form and then expanded to standard form ax^2 + bx + c = 0.

Given Data / Assumptions:

• Roots: r1 = 3 and r2 = −1.
• We are to write a monic quadratic (leading coefficient 1).
• Standard form required for comparison with options.

Concept / Approach:
If r1 and r2 are roots, the monic polynomial is (x − r1)(x − r2) = 0. Expand the product to get coefficients. Alternatively, use sum and product of roots: sum = r1 + r2, product = r1 * r2. Then x^2 − (sum)x + (product) = 0.

Step-by-Step Solution:

Verification / Alternative check:
Factor x^2 − 2x − 3 = (x − 3)(x + 1) = 0 which gives roots 3 and −1 as required.

Why Other Options Are Wrong:
Each incorrect option has either the wrong sign for x-term or constant term, yielding different roots upon factoring.

Common Pitfalls:
Mixing up signs when forming x^2 − (sum)x + (product); always compute sum and product carefully.

Final Answer:
x^2 − 2x − 3 = 0