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2002 AMC 10B Problem 4

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2002 AMC 10B Problem 4AMC 10 Step by Step

Name: Date: September 13, 2026

Algebraic Manipulation · difficulty 1 of 5 · about 1 min · position P1-10

What is the value of $(3x - 2)(4x + 1) - (3x - 2)4x + 1$ when $x=4$ ?

A)
$0$
B)
$1$
C)
$10$
D)
$11$
E)
$12$

Answer

Key insight

The two products share the factor (3x-2); pulling it out leaves (3x-2)(4x+1-4x) + 1 = 3x - 1, which is 11 when x = 4.

Solution

The first two terms both contain $(3x - 2)$ . Factor it out, being careful that the final $+1$ stands alone:
$(3x-2)(4x+1) - (3x-2)\cdot 4x + 1 = (3x-2)\big[(4x+1) - 4x\big] + 1 = (3x-2)\cdot 1 + 1 = 3x - 1.$

At $x = 4$ this is $3\cdot 4 - 1 = 11$ .

The answer is $\boxed{\textbf{(D)}\ 11}$ .

Why this works

When two terms share a factor, factoring is faster and safer than expanding: the bracket $(4x+1) - 4x$ collapses to $1$ , so the whole expression is linear. Parsing matters too: $-(3x-2)4x + 1$ means "subtract the product $(3x-2)\cdot 4x$ , then add $1$ ," and the $+1$ is not inside any product.

Alternative approach

Substitute $x = 4$ immediately: $(10)(17) - (10)(16) + 1 = 170 - 160 + 1 = 11$ .

The trap

Reading the trailing +1 as part of the second product, i.e. treating the expression as (3x-2)(4x+1) - (3x-2)(4x+1) = 0.

Common mistakes

Reading the trailing +1 as part of the second product, i.e. treating the expression as (3x-2)(4x+1) - (3x-2)(4x+1) = 0.
Dropping the $+1$ after factoring and answering $3x - 2 = 10$ , choice (C).

Techniques

Apply an identity: SFFT, sum of squares, difference of cubes, Vieta