To find the coordinates of the point on the curve where the tangent passes through the origin, we need to find the equation of the tangent line and then solve for the x-coordinate of the point of tangency.
Let's break it down step by step:
1. Start with the equation of the curve: y = x² + 3x + 4.
2. Find the derivative of the curve to find the slope of the tangent line. The derivative of y = x² + 3x + 4 is dy/dx = 2x + 3.
3. Set the derivative equal to the slope of the line passing through the origin, which is 0 since the line is horizontal. So, 2x + 3 = 0.
4. Solve the equation to find the x-coordinate of the point of tangency. Subtract 3 from both sides: 2x = -3. Divide by 2: x = -3/2.
5. Plug the x-coordinate back into the original equation to find the y-coordinate. y = (-3/2)² + 3(-3/2) + 4 = 9/4 - 9/2 + 4 = 9/4 - 18/4 + 16/4 = 7/4.
6. Therefore, the point on the curve where the tangent passes through the origin is (-3/2, 7/4).
None of the given options match the coordinates (-3/2, 7/4). Therefore, none of the options are correct.