What is the product of 2x2 − x + 1 and 3 − 2x

Analyze the question: We need to multiply the polynomial (2x² − x + 1) by (3 − 2x). Each term in the first bracket must be multiplied by each term in the second.

Concept: Use the distributive law: (a + b + c)(d + e) = ad + ae + bd + be + cd + ce. Then collect like terms.

Solution:

Multiply each term of (2x² − x + 1) by 3:

• 2x² × 3 = 6x²
• −x × 3 = −3x
• 1 × 3 = 3

Multiply each term of (2x² − x + 1) by −2x:

• 2x² × (−2x) = −4x³
• −x × (−2x) = +2x²
• 1 × (−2x) = −2x

Add everything together:

−4x³ + (6x² + 2x²) + (−3x − 2x) + 3

= −4x³ + 8x² − 5x + 3

Check with x = 1: (2 − 1 + 1)(3 − 2) = 2 × 1 = 2 Option B: −4 + 8 − 5 + 3 = 2 ✓

(Option A gives 4 − 8 + 5 + 3 = 4, which fails, so it is wrong.)

Answer: The product is −4x³ + 8x² − 5x + 3, which is option B.

Tip: When an expression has a negative term like (3 − 2x), watch your signs carefully. A quick way to check on exam day is to substitute x = 1 into the question and into each option. It takes only a few seconds and eliminates wrong options fast.