Where do I go with this, help?

Here is the question

A parabola has equation: y = ax^2 + bx + c.

It p***** through the origin, and the tangent  at (1,0) on it makes an
angle of 45 degrees with OX.
Find the actual equation on the parabola


So far i have

Since the parabola p***** through the origin then c = 0

so the equation is just y = ax^2 + bx

Also the tangent of the line at (1,0) is 1 (since tan 45 = 1)
i dont know if this is needed or not but the equation of this line is
therefore
y = x - 1

Now this is as far as I got.
This question comes at the end of an introduction to Differentiation
chapter
the answer in the book is y = x^2 + x
Ive tried to use the answer to work backwards but not getting anywhere