Question

simplify the following expression.\\((2x^2 - x + 3)(2x + 5)\\)\\(?x^3 + \square x^2 + \square x + \square\\)

Explanation:

Step1: Use distributive property (FOIL for polynomials)

Multiply each term in the first polynomial by each term in the second polynomial:

• \(2x^2 \times 2x = 4x^3\)
• \(2x^2 \times 5 = 10x^2\)
• \(-x \times 2x = -2x^2\)
• \(-x \times 5 = -5x\)
• \(3 \times 2x = 6x\)
• \(3 \times 5 = 15\)

Step2: Combine like terms

• For \(x^3\) term: Only \(4x^3\), so coefficient is 4.
• For \(x^2\) terms: \(10x^2 - 2x^2 = 8x^2\), so coefficient is 8.
• For \(x\) terms: \(-5x + 6x = x\), so coefficient is 1.
• Constant term: 15.

Answer:

The coefficient of \(x^3\) is 4, so the first box is filled with 4. The simplified polynomial is \(4x^3 + 8x^2 + x + 15\). So the answers for each box (in order) are 4, 8, 1, 15. But for the first box (the \(x^3\) coefficient), the answer is 4.