In algebraic factorization, completely factorize the cubic polynomial 48x^3 − 8x^2 − 93x − 45 into a product of three linear factors. Which of the following options gives the correct factorization?

Introduction / Context:
This algebra question tests your ability to factorize a cubic polynomial into linear factors. Factorization of higher degree polynomials is an important skill in algebra, as it helps in solving equations, simplifying expressions, and understanding polynomial graphs. Here, the polynomial 48x^3 − 8x^2 − 93x − 45 is already presented together with several candidate factorizations, and you must determine which product of three linear factors reproduces the given cubic exactly.

Given Data / Assumptions:

• The polynomial is P(x) = 48x^3 − 8x^2 − 93x − 45.
• We are given multiple options, each expressing P(x) as a product of three linear binomials.
• The task is to identify which option is the exact factorization of P(x).
• All arithmetic is over real numbers (the factors shown are real and rational).

Concept / Approach:
There are two main ways to tackle this problem. One way is to use the factor theorem by trying to find rational roots of P(x) using possible factors of the constant term and the leading coefficient. Another way, which fits well with the multiple choice structure, is to expand each candidate factorization and check which one matches the original polynomial term by term. Because the coefficients here are relatively small integers, expanding is manageable and allows us to verify the correct option systematically.

Step-by-Step Solution:
1) Consider option c: (4x + 3)(4x + 3)(3x − 5). 2) First multiply (4x + 3)(4x + 3) to obtain a quadratic factor: (4x + 3)^2 = 16x^2 + 24x + 9. 3) Now multiply this quadratic by (3x − 5): (16x^2 + 24x + 9)(3x − 5). 4) Distribute 3x across the quadratic: 3x * 16x^2 = 48x^3, 3x * 24x = 72x^2, and 3x * 9 = 27x. 5) Distribute −5 across the quadratic: −5 * 16x^2 = −80x^2, −5 * 24x = −120x, and −5 * 9 = −45. 6) Combine like terms: the x^3 term is 48x^3. The x^2 terms are 72x^2 − 80x^2 = −8x^2. The x terms are 27x − 120x = −93x. The constant term is −45. 7) The resulting polynomial is 48x^3 − 8x^2 − 93x − 45, which matches P(x) exactly.

Verification / Alternative check:
To be fully confident, we can briefly check one other option to see that it does not match. For instance, option a is (4x + 3)(4x − 3)(3x − 5). Multiplying (4x + 3)(4x − 3) first gives 16x^2 − 9. Multiplying by (3x − 5) then yields (16x^2 − 9)(3x − 5) = 48x^3 − 80x^2 − 27x + 45. This cubic clearly does not match the original because the x^2 and constant terms differ in both sign and value. Similar mismatches occur if you expand the other incorrect options, confirming that only option c matches P(x) exactly.

Why Other Options Are Wrong:
Option a produces 48x^3 − 80x^2 − 27x + 45, which differs in three coefficients. Option b has a repeated factor (4x − 3) and leads to a very different x^2 coefficient. Option d includes (3x + 5) as a factor, which gives a positive constant term when expanded, whereas P(x) has a negative constant term. Option e uses factors that generate degrees and coefficients inconsistent with 48x^3 − 8x^2 − 93x − 45. Only option c produces exactly the original polynomial upon expansion.

Common Pitfalls:
Learners sometimes try to factor by guessing only one factor, leading to time consuming algebra. Another common issue is making small arithmetic mistakes when expanding products, especially with negative signs and combined like terms. Working systematically, expanding one pair of factors at a time, and double checking coefficients for x^3, x^2, x, and the constant term help avoid such errors. In multiple choice settings, careful expansion of the most promising option is often enough to identify the correct factorization quickly.

Final Answer:
The correct complete factorization of 48x^3 − 8x^2 − 93x − 45 is (4x + 3)(4x + 3)(3x - 5).