Quadratic Division - How to divide two quadratics?

My studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula.

However, while I have been successful in understanding addition, subtraction and multiplication of 'g' and 'f':

Say

$f(x)=2{x}^{2}+15x-8$

$g(x)={x}^{2}+10x+16$

I have been unsuccessful in understanding division,

Where f(x)/g(x)

Would one start by grouping the functions into their own families?, for example:

$2{x}^{2}/{x}^{2}$

$15x/10x$

$-8/16$

I only came to this conclusion as with multiplication you times as so: $(A+B)\cdot (C+D)=AC+AD+BC+BD$

However from the example on Khan Ac. this seems wrong, it appears that division departs from any systematic way to find the answer, and that Khans division explanation is not as clear as his others on +, -, and * (appears to me!)

I decided to try and explore division once more in its most simple form to see if I was missing something.

$\frac{2}{3}=\mathrm{0.666666666...}$

$\frac{2}{4}=0.5$

Here clearly, certain division of numbers produces what seems like a simple solution into irrational numbers, If I were to divide a cake the same way I would think of each piece as a ratio, however in decimals these numbers are not clearly defined.

And when one starts dividing multiple factors as with above, how does one find the correct way to divide a quadratic equation, without relying on cheap tricks.

My studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula.

However, while I have been successful in understanding addition, subtraction and multiplication of 'g' and 'f':

Say

$f(x)=2{x}^{2}+15x-8$

$g(x)={x}^{2}+10x+16$

I have been unsuccessful in understanding division,

Where f(x)/g(x)

Would one start by grouping the functions into their own families?, for example:

$2{x}^{2}/{x}^{2}$

$15x/10x$

$-8/16$

I only came to this conclusion as with multiplication you times as so: $(A+B)\cdot (C+D)=AC+AD+BC+BD$

However from the example on Khan Ac. this seems wrong, it appears that division departs from any systematic way to find the answer, and that Khans division explanation is not as clear as his others on +, -, and * (appears to me!)

I decided to try and explore division once more in its most simple form to see if I was missing something.

$\frac{2}{3}=\mathrm{0.666666666...}$

$\frac{2}{4}=0.5$

Here clearly, certain division of numbers produces what seems like a simple solution into irrational numbers, If I were to divide a cake the same way I would think of each piece as a ratio, however in decimals these numbers are not clearly defined.

And when one starts dividing multiple factors as with above, how does one find the correct way to divide a quadratic equation, without relying on cheap tricks.