Every quadratic equation has two solutions; every quadratic expression can be factored. But nearly always when we are asked to factor a quadratic, we want to factor it "over the integers". And while every quadratic expression can be factored, very few can be factored over the integers.
So I think the answer to your question is that your quadratic expression can't be factored.
I assume you are familiar with the quadratic formula, and with the discriminant, b^2-4ac. A quadratic expression can be factored over the integers if and only if the discriminant is a perfect square (so that the square root of the discriminant is an integer).
In your quadratic, b^2-4ac is -15. So not only can the expression not be factored over the integers; but also the negative value of the discriminant means the zeros of the expression (the roots of the equation) are complex.
The quadratic formula gives the two roots of your example as
or
Then, if you wanted a factored form of your quadratic equation, even though the roots are complex, it would be
