To find the slope of the curve at a specific point, we need to find the derivative of the given function \(y = 2x^2 + 5x - 3\) and then evaluate it at the point \((1, 4)\).

The derivative of \(y\) with respect to \(x\) is given by:

\[\frac{dy}{dx} = \frac{d}{dx} (2x^2 + 5x - 3).\)

Using the power rule of differentiation, we get:

\[\frac{dy}{dx} = 4x + 5.\)

Now, we can evaluate the derivative at the point \((1, 4)\):

\[\frac{dy}{dx} \bigg|_{x=1} = 4(1) + 5 = 4 + 5 = 9.\)

Therefore, the slope of the curve at the point \((1, 4)\) is 9.