How do I find the solutions and factors of a function when given only the graph?

Use the graph to find the zeroes of the function. So, you find the points at which the function crosses the x-axis.

If f(x)=0 for x = x1, x = x2, x = x3, then:

f(x) = (x-x1)(x-x2)(x-x3) g(x)

where g(x) has no zeroes.