Question

How do you expand a algebraic expression?

Answer 1

Using the distributive property:

-if you have (or something of the same kind): a(b+c), this expands to ab+ac (1)
-if you have: (a+b)(c+d) this expands to ac+ad+bc+bd (2)

The last one is memorized by FOIL:
First: multiply first numbers of each factor: ac
Outer: multiply the outer numbers: ad
Inner: multiply the inner numbers: bc
Last: multiply the last numbers: bd

This is the starting point.
If you've got something more complex, always try to recognize these forms.

e.g.: expand 2x²(x+2)(x+4).

There are three main factors here: 2x², x+2 and x+4.
You can do multiplications in any order you like: 4*3*2 = (4*3)*2=4*(3*2)=(4*2)*3 etc.
Suppose you want to multiply 2x² and x+2 first:

2x²(x+2)=2x³+4x², using distributive property (1)

Now multiply 2x³+4x² and x+4, using the FOIL method (2):\[(2x^3+4x^2)(x+4)=2x^4+8x^3+4x^3+16x^2\]
This last expression can be simplified by adding like terms together (2nd and 3rd one):\[2x^4+12x^3+16x^2\]That's all there is to it!