Deductions from the Given Information

The given information provides us with a list of four numbers ($1$, $\sqrt{2}$, $x$, and $x^2$) and tells us that the range is $4$. So, we know the following:

$$the \: largest \: number - the \: smallest \: number = 4$$

But what's the largest number? What's the smallest number? If $x$ is less than $1$, say $\frac{1}{2}$, then $\sqrt{2}$ would be the largest number, $x^2 = \frac{1}{4}$ would be the smallest number, and the range would be $\sqrt{2} - \frac{1}{4}$. Clearly, this is NOT equal to $4$.

So $x$ must be greater than $1$ and the largest number must be $x^2$. That makes the smallest number $1$ and we can set up the following equation:

$$x^2 - 1 = 4$$

Solving the Problem

The deduction above is really the hard part of this problem. Once it is made, it's rather trivial. We just solve for $x$ in the following way:

$$x^2 - 1 = 4$$

$$x^2 = 5$$

$$x=\sqrt{5} \approx 2.24$$

$2.24$ is definitely larger than $2$, so the correct answer is A, Quantity A is larger.