How can you create a quadratic equation that cannot be solved by factoring but can be solved using the quadratic formula?

A quadratic equation can also have complex roots. In that
case too it would not be possible to arrive at the solution by factoring; instead we
would have to use the quadratic formula.

Complex roots of a
quadratic equation are always complex conjugates. If one root is a+ ib, the other is a -
ib.

The quadratic equation would
be:

(x - (a + ib))(x - (a - ib)) =
0

=> (x - a - ib)(x - a + ib) =
0

=> (x - a)^2 - (ib)^2 =
0

=> x^2 + a^2 - 2ax - i^2b^2 =
0

=> x^2 + a^2 - 2ax + b^2 =
0

=> x^2 - 2ax + a^2 + b^2 =
0

The equation we get is of the form x^2 -
2ax + a^2 + b^2 = 0