The expression b^2-4ac is called the discriminant. If it is zero, or positive and a perfect square (like 4, 9, 25, 36, etc), the solutions to the quadratic equation will be rational numbers and factoring will work.
On the other hand, if the discriminant is positive but not a perfect square, the solutions will be irrational, and if it is negative, there are no real solutions. In either of those cases, the quadratic expression cannot be factored.
In the equation 2x^2+7x+3=0, the discriminant is b^2-4ac = 7^2 - 4*2*3= 49-24 = 25 = 5^2
That means that the solutions can be found by factoring.
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