解这道数学题---The extracted content from the image is as follows: **Heading:** 解答题 (本题共 1 小题) (Problem Solving (This question has 1 sub-question)) **Question Stem (Problem 4):** 已知抛物线 y = x² + 2x + c 与 x 轴正半轴、y 轴负半轴分别交于点 A、B, 与 y 轴正半轴交于点 C, 且满足 OA = OB. (Given parabola y = x² + 2x + c intersects the positive x-axis at point A, the negative y-axis at point B, and the positive y-axis at point C, and satisfies OA = OB.) **Sub-questions:** (1) 求抛物线的解析式及顶点 H 的坐标; (Find the analytical expression of the parabola and the coordinates of its vertex H;) (2) 过y轴上一点 Q(0, q)作平行于 x 轴的水平直线交抛物线于 E, F 两点, 若线段 EF 长为 5, 求 q 的值; (Through a point Q(0, q) on the y-axis, draw a horizontal line parallel to the x-axis, intersecting the parabola at points E and F. If the length of segment EF is 5, find the value of q;) (3) 已知点 P(-3, 1), Q(2t, 2t+1), 且线段 PQ 与抛物线 y = x² + 2x + c 有且只有一个公共点, 直接写出 t 的取值范围. (Given point P(-3, 1), Q(2t, 2t+1), and segment PQ has exactly one common point with the parabola y = x² + 2x + c, directly write the range of t.) **Mathematical Formulas/Expressions:** * Parabola equation: `y = x² + 2x + c` * Length relationship: `OA = OB` * Point Q (in sub-question 2): `Q(0, q)` * Segment length: `EF = 5` * Point P (in sub-question 3): `P(-3, 1)` * Point Q (in sub-question 3): `Q(2t, 2t+1)` **Chart/Diagram Description:** * **Type:** Cartesian coordinate system with a parabola. * **Main Elements:** * **Coordinate Axes:** A horizontal axis labeled 'X' and a vertical axis labeled 'Y'. The origin is labeled 'O'. * **Parabola:** A curve that opens upwards, depicted in the coordinate system. * **Points:** * Point A: Located on the positive X-axis, representing an intersection point of the parabola with the x-axis. * Point C: Located on the positive Y-axis, representing an intersection point of the parabola with the y-axis. * Point B: Located on the negative Y-axis, representing an intersection point of the parabola with the y-axis. * Point H: The vertex of the parabola. * **Lines/Segments:** * The parabola is drawn passing through points A, C, and B. * A dashed vertical line extends downwards from the vertex H to the x-axis, indicating the axis of symmetry. * **Relative Position:** Point A is to the right of the origin O. Point C is above the origin O. Point B is below the origin O. The vertex H appears to be in the fourth quadrant.