Let's look at the example of (a+b)^2. The ^2 bit means "squared", or multiplied by itself. So it is basically just a fancy way of telling us to take two lots of (a+b) and multiply them together:
Expanding brackets means multiplying them together. It is important remember this as in the expression shown above, there isn't any sign between the brackets. Therefore, what we actually mean is
(a+b) x (a+b)
Similarly, when we multiply a by b, we write just ab to mean a x b. The reason why in algebra we do not show the multiplication sign is that mathematicians are lazy, and as multiplication is used so much in algebra, they can't be bothered to write "x" all the time, and it can also very easily be confused with the algebraic letter x, which is used very commonly in algebra.
So, now it's time to expand these brackets. When you multiply out brackets, you have to multiply every term in the first bracket with every term in the second bracket. This is shown below:
(a+b)(a+b) = a x a + ab +ba + b x b
We can now tidy this up a bit. On the outside, we have a x a and b x b. We can write these as a^2 and b^2. This is kind of the opposite of what we did right at the beginning. Also, ab = ba (you can try this with numbers - e.g. 7 x 8 = 8 x 7). So we have two lots of ab, which we can rewrite as 2ab. Our expansion becomes
(a+b)(a+b) = a^2 + 2ab + b^2
Excellent! We have successfully expanded (a+b)^2! The first part of the expansion that got us a x a + ab + ba + b x b may have been a bit hard to follow, or difficult to remember. There are various ways of remembering how to do this first part of the expansion (not all of them nice) but there are a couple civilised ways of doing it shown below:
Method 1: FOIL
This is a fun little mnemonic to help you learn how to expand brackets. It goes like this:
F stands for First - multiply the first term (letter) in each bracket together.
O stands for Outside - multiply the outside two terms together.
I stands for Inside - multiply the inside two terms together.
L stands for Last - multiply the last two terms in each bracket together.
Method 2: Multiplication Tables
If you thought you had rid yourself of those when you'd finally learnt all you times tables, then you were wrong! Even if you do maths at university level you find yourself doing them whilst studying Group Theory. No, you're not done with them yet. Times tables are for life, not just for junior school. Here we go then...
x, a, b
a a^2 ab
b ba b^2
Hopefully that didn't bring back too many painful memories. At least now, armed with an understanding of how to write algebraic expressions and expand brackets by a variety of different methods, you have at your fingertips of the most fundamental skills that gives you the potential to expand your mathematical knowledge in a variety of different areas!