Acurve has the slope 2x+3 at each point (x, y) on the curve. what is an equation for this curve if it passes through the point (1,2)

Therefore, f(x) = 2x + 3 is your derivative function, and you need to find the original curve. So find the antiderivative using the given conditions... 

∫f(x) = ∫2x + 3 dx 
F(x) = x^2 + 3x + C 
2 = (1)^2 + 3(1) + C 
2 = 4 + C 
C= -2 

Therefore, the curve is F(x) = x^2 + 3x - 2 

Proof: The derivative is the slope at every (x, y) point. The derivative of F(x) comes out to be 2x + 3, so we have found the curve. Plug in x = 1, and y = 2, so the conditions have been met. 

Hope I helped.

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