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?
 A: You have to multiply every term by every other term. A good way to make sure you don't miss any is to use a table.
First, combine like terms within each group of parentheses:
$$(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)\\
=(3x^4-7x^3+5x^2+5x+8)(-4x^2+3x+9)$$
Then form a table and multiply each term by multiplying the coefficients and adding the exponents:
$$
\begin{array}{c|cc}
\text{} & 3x^4 & -7x^3 & 5x^2 & 5x & 8 \\
\hline
-4x^2 & -12x^6 & -28x^5 & -20x^4 & -20x^3 & -32x^2 \\
3x & 9x^5 & -21x^4 & 15x^3 & 15x^2 & 24x \\
9 & 27x^4 & -63x^3 & 45x^2 & 45x & 72 \\
\end{array}
$$
Now take the new polynomial from the table and combine like terms:
$$-12x^6-28x^5-20x^4-20x^3-32x^2+9x^5-21x^4+15x^3\\
+15x^2+24x+27x^4-63x^3+45x^2+45x+72\\$$$$
=-12x^6+(9-28)x^5+(27-21-20)x^4+(-63+15-20)x^3\\+(45+15-32)x^2+(45+24)x+72\\$$$$
=-12x^6-19x^5-14x^4-68x^3+28x^2+69x+72$$
This method will also work with negative and non-integer exponents, as it is not restricted to polynomials.
A: 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?
