Expand and simplify (5x – 2y)(3x – 4y)

In this question, we are asked to 'expand' out the brackets. What this means is we need each term in one set of brackets (5x and -2y) to be multiplied by each term in the other set of brackets (3x and -4y).

So let's take this step-by-step.

We're going to use a simple technique which will help us in any future questions where we need to expand brackets. This is the FOIL technique:

F = first

O = outside

I  = inside

L = last

1) We start from F of FOIL by multiplying together both of our first terms.

These are 5x and 3x because if we look at each bracket separately, these are the two individual terms that come first:

5x*3x = 15x^2

(5 times 3 = 15 and x times x = x^2)

2) Next we use of FOIL by multiplying together the terms on the outside of the brackets. 

These are 5x and -4y because when both brackets are side by side, we can see that these terms are on the outer part of the equation.

5x*-4y = -20xy

(5 times -4 = -20 and x times y = xy)

3) Then we use of FOIL by multiplying together the terms on the inside of the brackets. 

These are -2y and 3x because when both brackets are side by side, we can see that these terms are on the inner part of the equation.

-2y*3x = -6xy

(-2 times 3 = -6 and x times y = xy)

4) Finally we use L of FOIL by multiplying together the last terms of the brackets. 

These are -2y and -4y because if we look at each bracket separately, these are the two individual terms that come last:

-2y*-4y = 8y^2

(-2 times -4 = 8 and y times y = y^2)

5) Now let's look at what terms we're left with after each of those 4 steps using our FOIL technique:

15x^2 - 20xy - 6xy + 8y^2

Our last step is to 'simplify' our answer by collecting any like terms. These are terms which have the same algebraic ending when we ignore the number in front of them. 

So in this case, we have xy terms which we can collect: -20xy-6xy = -26xy.

6) Our final answer will be 15x^2 - 26xy + 8y^2.


Do check your signs when you look over your answers and your working. It's easy to make silly mistakes when you multiply a negative term with another negative term, for example (the answer will be positive).

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