What am I doing wrong? (Simplifying Rational Expressions)

[eleventhxhour]

7a) Simplify and state any restrictions on the variables:

$$\frac{x^2-5xy+4y^2}{x^2+3xy-28y^2} ⋅ \frac{x^2+2xy+y^2}{x^2-y^2} $$

I'm not really sure what a good process would be to simplify this. This is what I tried to do, (below) which is wrong. Could anyone point out what I did wrong and what a better process to simplify it would be? Thanks!

My answer:

$$\frac{-5xy+4y^2}{3xy-28y^2} ⋅ \frac{2xy + y^2}{-y^2}$$

$$\frac{-10x^2y^2 + 4y^4}{-3xy^3+28y^4}$$

$$\frac{2y^2(-5x^2+2y^2}{y^3(-3x+28y)}$$

And then I'm not sure what to do.

 

[MarkFL]

I would suggest factoring first, and then from this you can see the restrictions and then simplify:

$$\frac{(x-y)(x-4y)}{(x+7y)(x-4y)}\cdot\frac{(x+y)^2}{(x+y)(x-y)}$$

 

[soroban]

Hello, eleventhxhour!

What are you doing wrong? . . . Everything!

Simplify: .$$\frac{x^2-5xy+4y^2}{x^2+3xy-28y^2} ⋅ \frac{x^2+2xy+y^2}{x^2-y^2} $$

You have: .$$\frac{x^2-5xy+4y^2}{x^2+3xy-28y^2} ⋅ \frac{x^2+2xy+y^2}{x^2-y^2} $$

Then you canceled illegally:

. . $$\frac{{\color{red}\rlap{//}}x^2-5xy+4y^2}{{\color{red}\rlap{//}}x^2+3xy-28y^2} ⋅ \frac{{\color{red}\rlap{//}}x^2+2xy+y^2}{{\color{red}\rlap{//}}x^2-y^2} $$

And got: .[math]\frac{-5xy+4y^2}{+3xy-28y^2} ⋅ \frac{+2xy+y^2}{-y^2} [/math]Then you multiplied incorrectly.

You said: .[math](-5xy + 4y^2)(2xy + y^2) \:=\:-10x^2y + 4y^4[/math]

. . as if you never heard of "FOIL".