Question

what is the product?\\((-2a^2 + s)(5a^2 - 6s)\\)\\(\bigcirc -10a^4 + 17a^2s - 6s^2\\)\\(\bigcirc -10a^4 + 17a^4s^2 - 6s^2\\)\\(\bigcirc -10a^4 - 7a^2s - 6s^2\\)\\(\bigcirc -10a^4 + 17a^2s + 6s^2\\)

Explanation:

Step1: Apply the distributive property (FOIL method)

Multiply each term in the first binomial by each term in the second binomial:
\[

$$\begin{align*} (-2a^{2}+s)(5a^{2}-6s)&=-2a^{2}\times5a^{2}+(-2a^{2})\times(-6s)+s\times5a^{2}+s\times(-6s)\\ \end{align*}$$

\]

Step2: Simplify each term

• For \(-2a^{2}\times5a^{2}\): Using the rule \(a^{m}\times a^{n}=a^{m + n}\), we get \(-10a^{4}\).
• For \((-2a^{2})\times(-6s)\): Multiplying the coefficients and keeping the variables, we get \(12a^{2}s\).
• For \(s\times5a^{2}\): This simplifies to \(5a^{2}s\).
• For \(s\times(-6s)\): Using the rule \(a^{m}\times a^{n}=a^{m + n}\) (here \(a = s\), \(m = 1\), \(n = 1\)), we get \(-6s^{2}\).

So now the expression is:
\[
-10a^{4}+12a^{2}s + 5a^{2}s-6s^{2}
\]

Step3: Combine like terms

Combine the \(a^{2}s\) terms: \(12a^{2}s+5a^{2}s = 17a^{2}s\).

Putting it all together, we have:
\[
-10a^{4}+17a^{2}s-6s^{2}
\]

Answer:

\(-10a^{4}+17a^{2}s - 6s^{2}\) (the first option: \(-10a^{4}+17a^{2}s - 6s^{2}\))