Big Fact: Multiplying polynomials relies on the distributive property.
Example
$(7v)(-3v^2-8v+5)$
$$
\begin{array}{lll}
\displaystyle (7v)(-3v^2-8v+5)&\displaystyle=(7v)(-3v^2)-(7v)(8v)+(7v)(5) &\mbox{distribute}\\
\displaystyle &\displaystyle=-21v^3-56v^2+35v &\mbox{exponent rules}\\
\end{array}
$$
Example
$(g+2)(g-3)$
$$
\begin{array}{lll}
\displaystyle \color{blue}{(g+2)}\color{magenta}{(g-3)}&\displaystyle=\color{blue}{(g+2)}\color{magenta}{g}-\color{blue}{(g+2)}\cdot \color{magenta}{3}&\mbox{distribute from left}\\
\displaystyle &\displaystyle= \color{blue}{g}\cdot \color{magenta}{g}+\color{blue}{2}\cdot \color{magenta}{g}-\color{blue}{g}\cdot\color{magenta}{3}-\color{blue}{2}\cdot \color{magenta}{3}&\mbox{distribute from right}\\
\displaystyle &\displaystyle= g^2+2g-3g-6&\mbox{simplify}\\
\displaystyle &\displaystyle= g^2-g-6 & \mbox{combine like terms}\\
\end{array}
$$
Scenic Alternative: FOIL (First, Outer, Inner, Last). $$ \begin{array}{lll} \displaystyle (g+2)(g-3)&\displaystyle=\overbrace{g^2}^{\mbox{ First }}\overbrace{-3g}^{\mbox{ Outer}}\overbrace{+2g}^{\mbox{ Inner }}-\overbrace{2\cdot 3}^{\mbox{ Last }}&\mbox{using FOIL shortcut}\\ \displaystyle &\displaystyle= g^2-g-6&\mbox{combine like terms}\\ \end{array} $$
Scenic Alternative: FOIL (First, Outer, Inner, Last). $$ \begin{array}{lll} \displaystyle (g+2)(g-3)&\displaystyle=\overbrace{g^2}^{\mbox{ First }}\overbrace{-3g}^{\mbox{ Outer}}\overbrace{+2g}^{\mbox{ Inner }}-\overbrace{2\cdot 3}^{\mbox{ Last }}&\mbox{using FOIL shortcut}\\ \displaystyle &\displaystyle= g^2-g-6&\mbox{combine like terms}\\ \end{array} $$
Example
$(-2b^2+6b-8)(9b^2-3b)$
Here, we will distribute like crazy.
$$
\begin{array}{lll}
\displaystyle \color{blue}{(-2b^2+6b-8)}\color{magenta}{(9b^2-3b)}&\displaystyle=\color{magenta}{\color{blue}{(-2b^2+6b-8)}(9b^2)-\color{blue}{(-2b^2+6b-8)}(3b)} &\mbox{distribute from left}\\
\displaystyle &\displaystyle=\color{blue}{-2b^2\color{magenta}{(9b^2)}+6b\color{magenta}{(9b^2)}-8\color{magenta}{(9b^2)}}-\color{blue}{(-2b^2\color{magenta}{(3b)}+6b\color{magenta}{(3b)}-8\color{magenta}{(3b)})} &\mbox{distribute from right}\\
\displaystyle &\displaystyle=-18b^4+54b^3-72b^2-(-6b^3+18b^2-24b) &\mbox{simplify}\\
\displaystyle &\displaystyle=-18b^4+54b^3-72b^2+6b^3-18b^2+24b &\mbox{subtract (distribute minus!)}\\
\displaystyle &\displaystyle=-18b^4+60b^3-90b^2+24b &\mbox{combine like terms}\\
\end{array}
$$
Scenic Alternative: Box Method Shortcut
Begin by writing the terms (including signs!) in a box format. $$ \begin{array}{|c|c|c|} \hline & \color{magenta}{9b^2}&\color{magenta}{-3b}\\\hline \color{blue}{-2b^2} & &\\\hline \color{blue}{+6b} & &\\\hline \color{blue}{-8} & &\\\hline \end{array} $$ Filling in the boxes we have...
$$ \begin{array}{|c|c|c|} \hline & \color{magenta}{9b^2}&\color{magenta}{-3b}\\\hline \color{blue}{-2b^2} & -18b^4 & \color{darkorange}{+6b^3}\\\hline \color{blue}{+6b} & \color{darkorange}{+54b^3}&\color{red}{-18b^2}\\\hline \color{blue}{-8} & \color{red}{-72b^2}&+24b\\\hline \end{array} $$ Thus, with the above $$ \begin{array}{lll} \displaystyle \color{blue}{(-2b^2+6b-8)}\color{magenta}{(9b^2-3b)}&\displaystyle=-18b^4\color{darkorange}{+54b^3+6b^3}\color{red}{-72b^2-18b^2}+24b &\mbox{}\\ \displaystyle &\displaystyle=-18b^4+60b^3-90b^2+24b &\mbox{combine like terms}\\ \end{array} $$ just like the answer above!
Scenic Alternative: Box Method Shortcut
Begin by writing the terms (including signs!) in a box format. $$ \begin{array}{|c|c|c|} \hline & \color{magenta}{9b^2}&\color{magenta}{-3b}\\\hline \color{blue}{-2b^2} & &\\\hline \color{blue}{+6b} & &\\\hline \color{blue}{-8} & &\\\hline \end{array} $$ Filling in the boxes we have...
$$ \begin{array}{|c|c|c|} \hline & \color{magenta}{9b^2}&\color{magenta}{-3b}\\\hline \color{blue}{-2b^2} & -18b^4 & \color{darkorange}{+6b^3}\\\hline \color{blue}{+6b} & \color{darkorange}{+54b^3}&\color{red}{-18b^2}\\\hline \color{blue}{-8} & \color{red}{-72b^2}&+24b\\\hline \end{array} $$ Thus, with the above $$ \begin{array}{lll} \displaystyle \color{blue}{(-2b^2+6b-8)}\color{magenta}{(9b^2-3b)}&\displaystyle=-18b^4\color{darkorange}{+54b^3+6b^3}\color{red}{-72b^2-18b^2}+24b &\mbox{}\\ \displaystyle &\displaystyle=-18b^4+60b^3-90b^2+24b &\mbox{combine like terms}\\ \end{array} $$ just like the answer above!
Example
Use the box method to multiply the polynomials $$(-5y^2-9y+6)(3y^2-8y-6)$$
First, we make the boxes.
$$
\begin{array}{|c|c|c|c|}
\hline
& 3y^2 &-8y & -6 \\\hline
-5y^2 & -15y^4 & +40y^3 & +30y^2\\\hline
-9y & -27y^3 & +72y^2&+54y\\\hline
+6 & 18y^2 & -48y&-36\\\hline
\end{array}
$$
Noticing that like terms are on the diagonals, we perform the multiplication.
$$
\begin{array}{lll}
\displaystyle (-5y^2-9y+6)(3y^2-8y-6)&\displaystyle= -15y^4-27y^3+40y^3+18y^2+72y^2+30y^2-48y+54y-36&\mbox{}\\
\displaystyle &\displaystyle=-15y^4+13y^3+120y^2+6y-36 &\mbox{}\\
\end{array}
$$