Question

find the product.\\((2x^4 + 3x^2)(x^4 - x^2)\\)\\(?x^8 + \quadx^6 + \quadx^4\\)

Explanation:

Step1: Apply distributive property (FOIL)

Multiply \(2x^4\) with each term in \((x^4 - x^2)\) and \(3x^2\) with each term in \((x^4 - x^2)\):
\(2x^4 \cdot x^4 - 2x^4 \cdot x^2 + 3x^2 \cdot x^4 - 3x^2 \cdot x^2\)

Step2: Use exponent rule \(a^m \cdot a^n = a^{m + n}\)

Simplify each term:
\(2x^{4 + 4} - 2x^{4 + 2} + 3x^{2 + 4} - 3x^{2 + 2}\)
\(2x^8 - 2x^6 + 3x^6 - 3x^4\)

Step3: Combine like terms

Combine the \(x^6\) terms:
\(2x^8 + (-2x^6 + 3x^6) - 3x^4\)
\(2x^8 + x^6 - 3x^4\)

Answer:

For the \(x^8\) term: \(2\)
For the \(x^6\) term: \(1\)
For the \(x^4\) term: \(-3\)

(If we only consider the coefficient for \(x^8\) as per the green box, the answer is \(2\))