Question

How do you multiply `(x + -1y)(x + y)`?

Answer 1

`x^2-y^2`

Explanation:

`color(blue)("Using the shortcut with a bit of explanation")`

It is the case that `1y` is written as just `y`

We also have `+ -`. When two signs are next to each other and they are different the result is -. So `+ - 1y -> -y` giving

`(x+ - 1y)(x+y)" "->" "(x-y)(x+y)`

The shortcut is to know that if you have the form `a^2-b^2` then this works out to be the same as `(a-b)(a+b)`

That is the condition in this question so `(x-y)(x+y)=x^2-y^2`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Demonstration that this is true")`

Consider:`" "color(blue)((x-y))color(brown)((x+y))`

Multiply everything inside the right hand side bracket by everything inside the left hand side bracket.

`color(brown)(color(blue)(x)(x+y) color(blue)(-y)(x+y)`

Notice that the minus follow the `color(blue)(y)` in `color(blue)(-y)`

So we have:
`color(brown)(color(blue)(x)(x+y)" " color(blue)(-y)(x+y)`
`x^2+xy" "-xy -y^2`

`x^2+0-y^2`

`x^2-y^2`

Answer 2

`x^2-y^2`

Explanation:

Each term in the 2nd bracket must be multiplied by each term in the 1st bracket.

That is `(color(red)(x-y))(x+y)=color(red)(x)(x+y)color(red)(-y)(x+y)`

now distribute the brackets

`=x^2+cancel(xy)-cancel(xy)-y^2=x^2-y^2`