Question

express the product of (2x^2 + 6x - 8) and (x + 3) in standard form.

Explanation:

Step1: Identify the problem type

We need to find the product of two algebraic expressions: \(2x^2 + 6x - 8\) and \(x + 3\). This is a problem of multiplying polynomials, which falls under Algebra.

Step2: Use the distributive property (FOIL for binomials, extended for trinomials)

We multiply each term in the first polynomial by each term in the second polynomial:

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Step3: Simplify each term

• \(2x^2 \cdot x = 2x^{2 + 1} = 2x^3\)
• \(2x^2 \cdot 3 = 6x^2\)
• \(6x \cdot x = 6x^{1 + 1} = 6x^2\)
• \(6x \cdot 3 = 18x\)
• \(-8 \cdot x = -8x\)
• \(-8 \cdot 3 = -24\)

Step4: Combine like terms

Now we combine the terms with the same power of \(x\):

• For \(x^3\): \(2x^3\) (only term)
• For \(x^2\): \(6x^2 + 6x^2 = 12x^2\)
• For \(x\): \(18x - 8x = 10x\)
• Constant term: \(-24\) (only term)

Putting it all together, we get:

$$ 2x^3 + 12x^2 + 10x - 24 $$

Answer:

The product in standard form is \(2x^3 + 12x^2 + 10x - 24\)