Strategy: There are two possible approaches here. We can test each answer choice by plugging it into the given equation to see if it satisfies that equation.
Since testing answer choice B (-1/2) looks like a pain, I'm going to solve the given equation for x.

Given: $2x^4 = 7x^3 + 4x^2$
Since there appears to be a quadratic equation hiding in this equation, let's first set it equal to zero: $2x^4 - 7x^3 - 4x^2 = 0$
Factor out $x^2$ to get: $x^2(2x^2 - 7x - 4) = 0$

Since the first term of the quadratic equation inside the brackets is $2x^2$, we know that the expression will factor into something that looks like this: $x^2(2x ± ?)(x ± ?) = 0$

Notice that the last two remaining terms must multiply to get $-4$, so we need only test a few pairs of values until we got something that expands to get $2x^2 - 7x - 4 = 0$

Eventually we'll get: $x^2(2x + 1)(x - 4) = 0$
So it could be the case that $x^2 = 0$, which means $x = 0$
Or it could be the case that $2x+1 = 0$, which means $x = -1/2$
Or it could be the case that $x - 4 = 0$, which means $x = 4$

Answer: A, B and E