Let's use the strategy of matching operations to get one quantity equal to zero (I find it much easier to compare quantities when one of those quantities is zero)

Given:
Quantity A: $x^2-5x-14$
Quantity B: $2x^2-11x-4$

Subtract $x^2$ from both quantities to get:
Quantity A: $-5x-14$
Quantity B: $x^2-11x-4$

Add $5x$ to both quantities to get:
Quantity A: $-14$
Quantity B: $x^2-6x-4$

Add $14$ to both quantities to get:
Quantity A: $0$
Quantity B: $x^2-6x+10$

At this point I recognize the quantity B looks quite similar to a perfect square.
That is, I know that $x^2-6x+9$ can be factored to get $(x-3)^2$

So if we subtract 1 from both quantities something nice happens with Quantity B.
We get:
Quantity A: $-1$
Quantity B: $x^2-6x+9$

In other words:
Quantity A: $-1$
Quantity B: $(x-3)^2$

Since $(x-3)^2$ is greater than or equal to zero for ALL values of x, we can be certain that Quantity B is greater than Quantity A

Answer: B

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