Question

simplify the following expression.
$(2x^2 - 2x + 5)(x + 4)$
$?x^3 + \square x^2 + \square x + \square$

Explanation:

Step1: Apply distributive property (FOIL for polynomials)

Multiply each term in the first polynomial by each term in the second polynomial:
$$(2x^2 - 2x + 5)(x + 4) = 2x^2 \cdot x + 2x^2 \cdot 4 - 2x \cdot x - 2x \cdot 4 + 5 \cdot x + 5 \cdot 4$$

Step2: Simplify each product

Calculate each term:
$$2x^3 + 8x^2 - 2x^2 - 8x + 5x + 20$$

Step3: Combine like terms

Combine the \(x^2\) terms and the \(x\) terms:

• For \(x^2\): \(8x^2 - 2x^2 = 6x^2\)
• For \(x\): \(-8x + 5x = -3x\)

So the simplified expression is:
$$2x^3 + 6x^2 - 3x + 20$$

Answer:

The coefficient of \(x^3\) is \(2\), the coefficient of \(x^2\) is \(6\), the coefficient of \(x\) is \(-3\), and the constant term is \(20\). So filling in the blanks: \(2\) (for \(x^3\)), \(6\) (for \(x^2\)), \(-3\) (for \(x\)), and \(20\) (the constant).