First, let's write this as a quadratic equation and solve.
2x² + x - 3 = 0
(2x + 3)(x - 1) = 0
2x + 3 = 0
2x = -3
x = -3/2
and
x - 1 = 0
x = 1
Now, we need to examine the behavior of this quadratic at three different intervals:
(-∞, -3/2), (-3/2, 1), and (1, ∞)
Let's choose a number in each of the three intervals and observe the behavior of the factored quadratic and whether it satisfies the original inequality. The original inequality says that the expression on the left must be positive.
For the first interval, if we let x = -2 and plug it into the factored expression, then we have a negative number times a negative number, which yields a positive number.
For the second interval, let's choose x = 0. Then, substitution into the factored expression gives us a positive number times a negative number, which is a negative number.
For the third interval, let's select x = 2. Once we substitute into the factored expression, we obtain a positive number times a positive number, which clearly returns a positive number.
Therefore, the two intervals which satisfy the original inequality are:
(-∞, 3/2) and (1, ∞) or (-∞, 3/2) ∪ (1, ∞)
