Step by step
The question
Question: expand (a − 5)(a + 2)(2a − 1).
Expand means: multiply out all the brackets.
1 Multiply the first two
First the first two brackets: (a − 5)(a + 2).
The third waits.
Careful!
We can't just multiply the first parts and the last parts.
Every part meets every part.
Each part of the first bracket multiplies the whole second bracket.
The parts of the first bracket are a and −5.
Each one takes the whole second bracket.
What is in front of a bracket multiplies every part inside.
So we write each part times what is in front.
Watch the signs.
a times a is a².
a times 2 is 2a.
(−5) times a is −5a.
(−5) times 2 is −10.
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the a terms go together.
Each colour is one kind.
2a − 5a is −3a.
It is like 2 apples take away 5 apples: that is 3 apples.
So the expression is a² − 3a − 10.
2 Times the third bracket
Now this answer times the third bracket, (2a − 1).
Each part of the long bracket meets every part of the third.
Each part of the first bracket multiplies the whole second bracket.
The parts of the first bracket are a² and −3a and −10.
Each one takes the whole second bracket.
What is in front of a bracket multiplies every part inside.
So we write each part times what is in front.
Watch the signs.
a² times 2a is 2a³.
a² times (−1) is −a².
(−3a) times 2a is −6a².
(−3a) times (−1) is 3a.
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the a² terms go together.
Each colour is one kind.
−a² − 6a² is −7a².
3a − 20a is −17a.
It is like 3 apples take away 20 apples: that is 17 apples.
So the expression is 2a³ − 7a² − 17a + 10.
4 Check
Let's check.
We pick a number: a = 2.
We put it into the start: −36.
Now into our answer.
Both give −36.
Same value, so 2a³ − 7a² − 17a + 10 is right!
Let's look back.
First: multiply the first two.
Then: times the third bracket.
Then: group like terms.
Last: check.