Here's another approach:

Given: \(x^2=y^2+1\)

Square both sides to get: \((x^2)^2=(y^2+1)^2\)

Simplify both sides to get: \(x^4=y^4+2y^2+1\)

We'll use this information shortly

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We'll start with:
Quantity A: \(x^4\)
Quantity B: \(-y^4+1\)


Replace \(x^4\) with \(y^4+2y^2+1\) to get:
Quantity A: \(y^4+2y^2+1\)
Quantity B: \(-y^4+1\)

Subtract 1 from both quantities to get:
Quantity A: \(y^4+2y^2\)
Quantity B: \(-y^4\)

Subtract add \(y^4\) to both quantities to get:
Quantity A: \(2y^4+2y^2\)
Quantity B: \(0\)

Factor Quantity A to get:
Quantity A: \(2(y^4+y^2)\)
Quantity B: \(0\)

Since \(y \neq 0\), we know that \(y^4\) is POSITIVE and \(y^2\) is POSITIVE
So, \(y^4+y^2\) is POSITIVE

We get:
Quantity A: 2(some positive number)
Quantity B: 0

Clearly Quantity A is greater.

Answer: A

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