Question

multiply. (simplify your answer completely.)
(5n^{4}-4n^{2})(2n^{4}-n^{2})

Explanation:

Step1: Apply Distributive Property

Multiply \(5n^4 - 4n^2\) with each term in \(2n^4 - n^2\) separately.
\[

$$\begin{align*} &(5n^4 - 4n^2)(2n^4 - n^2)\\ =& 5n^4 \times 2n^4 + 5n^4 \times (-n^2) - 4n^2 \times 2n^4 - 4n^2 \times (-n^2) \end{align*}$$

\]

Step2: Multiply Coefficients and Exponents

For each term, multiply the coefficients and add the exponents of like bases (using \(a^m \times a^n = a^{m + n}\)).
\[

$$\begin{align*} &5 \times 2 \times n^{4 + 4}+5 \times (-1) \times n^{4 + 2}-4 \times 2 \times n^{2 + 4}-4 \times (-1) \times n^{2 + 2}\\ =& 10n^8 - 5n^6 - 8n^6 + 4n^4 \end{align*}$$

\]

Step3: Combine Like Terms

Combine the terms with \(n^6\).
\[
10n^8 + (-5n^6 - 8n^6) + 4n^4 = 10n^8 - 13n^6 + 4n^4
\]

Answer:

\(10n^8 - 13n^6 + 4n^4\)