Algebraic expansion of (x + 3)(x - 1): when this product is fully expanded and simplified, what quadratic expression in the standard form ax^2 + bx + c is obtained?

Introduction / Context:
This question tests basic algebraic expansion and simplification of a product of two linear binomials. Such problems are common in aptitude exams because they check whether you are comfortable with multiplying brackets and collecting like terms to reach a standard quadratic expression of the form ax^2 + bx + c.

Given Data / Assumptions:

• The expression to expand is (x + 3)(x - 1).
• We must express the result in standard quadratic form ax^2 + bx + c.
• No specific value of x is given; we work symbolically.

Concept / Approach:
To expand (x + 3)(x - 1), we use the distributive law of multiplication over addition. Each term in the first bracket is multiplied by each term in the second bracket. Then we combine like terms. The coefficient of x^2 gives a, the coefficient of x gives b, and the constant term gives c in the final quadratic expression.

Step-by-Step Solution:
First, multiply x by each term in the second bracket: x * x = x^2 and x * (-1) = -x.Next, multiply 3 by each term in the second bracket: 3 * x = 3x and 3 * (-1) = -3.Combine all four products: x^2 - x + 3x - 3.Combine like terms in x: -x + 3x = 2x, so the expression becomes x^2 + 2x - 3.Thus the standard quadratic form is x^2 + 2x - 3, where a = 1, b = 2, and c = -3.

Verification / Alternative check:
We can verify the expansion by substituting a simple value for x in both the original and expanded forms. For example, if x = 2, then (2 + 3)(2 - 1) = 5 * 1 = 5. Using the expanded form, x^2 + 2x - 3 becomes 2^2 + 2 * 2 - 3 = 4 + 4 - 3 = 5, which matches. This confirms that the expansion is correct.

Why Other Options Are Wrong:

• x^2 - 2x + 3 changes the signs of both the x term and the constant term, so it does not match the actual expansion.
• x^2 + 3x + 2 would come from (x + 1)(x + 2), not from (x + 3)(x - 1).
• x^2 - 4x + 3 corresponds to a different pair of binomials and does not give the same products.
• x^2 + 4x - 3 has an incorrect coefficient on x compared with the correct 2x term.

Common Pitfalls:

• Forgetting one of the cross terms, such as 3x or -x, which changes the middle coefficient.
• Making sign errors when multiplying by negative numbers, especially the -1 term.
• Incorrectly rearranging or simplifying like terms, leading to a wrong coefficient of x.

Final Answer:
x^2 + 2x - 3