Is my calculus book wrong

I'm differentiating x^2 + xy - y^2 = 4. When I differentiate, I get:
First, 2x + (x)dy/dx + y - (2y)dy/dx = 0
Next, 2x + y - (2y + x)dy/dx = 0
Then, (-2y + x)dy/dx = -2x + y
Solution: 2x + y / 2y + x (negatives cancel)

But, the book says that it's it's 2x + y / 2y - x. Is the book wrong, or is there a sign error that I may have done?

Book is right. You did a small mistake.

first step is right

2x + xdy/dx + y - 2y dy/dx = 0

2x + y - dy/dx(2y - x) = 0 (after taking - dy/dx as common factor x sign changes to negative)

2x + y = dy/dx(2y - x)

dy/dx = (2x + y)/(2y - x)

When going from the first step to the second, you thought you were factoring out dy/dx, when you were actually factoring out - dy/dx
So The second step should be 2x + y - (2y - x)dy/dx (if you're not convinced, expand this and see if you get 2x + y -2x dy/dx + xdy/dx)

Then the third line is full of sign mistakes!
it should be -(2y - x) dy/dx = -2x - y
So dy/dx = (-2x - y) / (-(2y-x))
You can take the - from the denominator to the numerator, so it becomes -(-2x-y) = 2x+y
SO the answer is dy/dx = (2x + y) / (2y - x)

You mis-canceled the negatives.

(-2x + y) turns into -(2x - y) and
(-2y + x) turns into -(2y - x)

because you need to flip *both* signs in the expression.

So when you divide one by the other you can cancel the negatives and get (2x - y)/(2y - x) so there must be something wrong somewhere else in your work.

During the "Next" step, the x within the parentheses should be negative :)

you should have subtracted (x)dy/dx and added (2y)dy/dx separately to get (2y-x)dy/dx=2x+y

you can do the math from there

Never trust statistics unless you fiddled the figures yourself.

... (x) dy/dx - (2y) dy/dx = 0

... - (2y - x) dy/dx = 0