Hence state the number of the solution in this domain of the equation x^2=3x+sinx?

hence
adverb
1.
as a consequence; for this reason.
"a stiff breeze and hence a high windchill"
synonyms: in consequence, consequently, as a consequence, for this reason, therefore, thus, so, accordingly, as a result, because of that, that being so, that being the case, on that account; ergo
"many vehicle journeys (and hence a lot of pollution) would be saved"
2.
in the future (used after a period of time).
"two years hence they might say something quite different"
~ google.com
So what are you talking about?

Domain is the collection of valid values for x. Domain appears to be "all real numbers."

To draw a diagram requires two unknowns. x^2=3x+sinx has only one. You could rewrite it as y = -x^2 + 3x +sin(x) and then draw a diagram of that. It looks like this:
www.wolframalpha.com/input/?i=plot+y+%3D+-x%5E2+%2B+3x+%2Bsin(x)

The combined diagram looks like this:
www.wolframalpha.com/input/?i=plot+y+%3D+-x%5E2+%2B+3x+%2Bsin(x),+y%3D3x-x%5E2,+y%3D-sinx

It is not clear just what you are trying to calculate. Math is only valid when it describes reality.