<s>[INST] How to multiply polynominals I can't figure out how to multiply these polynominals $$(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)$$
I tried multiplying like this
$$(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)$$
$$3x^4-7x^3+(5x^2)(2x^2)(-6x^2)+(5x)(7x)(-4x)+(8)(9)$$
$$3x^4-7x^3-60x^2-140x+72$$
It says the answer is $$-12x^6-19x^5-14x^4-68x^3+28x^2+69x+72$$ but how did they get it?
 [/INST] To multiply two polynomials, distribute each term in the left set of parentheses over the entire collection of terms in the right set of parentheses. Your example would start out like:
$$
\begin{align*}
& (\color{red}{5x^2+3x^4-7x^3+5x+8})(2x^2-4x+9-6x^2+7x)\\
= & \color{red}{5x^2}(2x^2-4x+9-6x^2+7x)\\
& \color{red}{+ 3x^4}(2x^2-4x+9-6x^2+7x)\\
& \color{red}{- 7x^3}(2x^2-4x+9-6x^2+7x)\\
& \color{red}{+ 5x}(2x^2-4x+9-6x^2+7x)\\
& \color{red}{+ 8}(2x^2-4x+9-6x^2+7x).
\end{align*}
$$
Can you take it from here?
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