Question: Expand the product $ (2x - 3)(x + 4)(x - 1) $.

February 22, 2026

Expanding the Product: $ (2x - 3)(x + 4)(x - 1) $

If you're working with cubic expressions in algebra, expanding products like $ (2x - 3)(x + 4)(x - 1) $ may seem tricky at first—but with the right approach, it becomes a smooth process. In this article, we’ll walk step-by-step through expanding the expression $ (2x - 3)(x + 4)(x - 1) $, explain key algebraic concepts, and highlight how mastering this technique improves your overall math proficiency.

Why Expand Algebraic Expressions?

Expanding products helps simplify expressions, solve equations, and prepare for higher-level math such as calculus and polynomial factoring. Being able to expand $ (2x - 3)(x + 4)(x - 1) $ not only aids in solving expressions but also strengthens problem-solving skills.

Step-by-Step Expansion

Step 1: Multiply the first two binomials

Start by multiplying $ (2x - 3) $ and $ (x + 4) $:

$$ (2x - 3)(x + 4) = 2x(x) + 2x(4) - 3(x) - 3(4) $$

$$ = 2x^2 + 8x - 3x - 12 $$

$$ = 2x^2 + 5x - 12 $$

Step 2: Multiply the result by the third binomial

Now multiply $ (2x^2 + 5x - 12)(x - 1) $:

Use the distributive property (also known as FOIL for binomials extended to polynomials):

$$ (2x^2 + 5x - 12)(x - 1) = 2x^2(x) + 2x^2(-1) + 5x(x) + 5x(-1) -12(x) -12(-1) $$

$$ = 2x^3 - 2x^2 + 5x^2 - 5x - 12x + 12 $$

Step 3: Combine like terms

Now combine terms with the same degree:

$ 2x^3 $
$ (-2x^2 + 5x^2) = 3x^2 $
$ (-5x - 12x) = -17x $
Constant: $ +12 $

So, the fully expanded expression is:

$$ oxed{2x^3 + 3x^2 - 17x + 12} $$

Tips for Efficient Expansion

Start with the simplest products first: Multiply two binomials before involving the third.
Use the distributive property carefully: Each term in the first polynomial must multiply by each term in the second.
Group terms by degree: This helps identify like terms and simplifies the final form.
Double-check signs and coefficients: A small error in signs or multiplications can change the expression entirely.

Real-World Applications

Understanding expansion helps in modeling real-world situations such as:

Calculating volumes or areas involving variables
Analyzing profit functions in economics
Solving physics problems involving polynomial relationships
Preparing for advanced algebra, engineering, and data science foundations

Frequently Asked Questions (FAQ)

Q: Why must I expand $ (2x - 3)(x + 4)(x - 1) $? A: Expanding removes parentheses, simplifies expressions, and is essential for solving equations or analyzing function behavior.

Q: Can I use a calculator to expand this? A: Yes, but manual expansion builds deeper understanding and precision, crucial for math proficiency.

Q: What happens if I make a sign error? A: A mistake in a negative coefficient can drastically change the result—always verify each step.

Conclusion

Expanding $ (2x - 3)(x + 4)(x - 1) $ follows standard algebraic rules and results in: