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How to find the equation of a parabola if it's focus and the equation of it's tangent passing through the vertex is given.

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The line connecting the focus and the vertex will be perpendicular to the line tangent at the vertex.       Since the tangent line at the vertex has slope 2  (because you can write it as y = 2x+4),  that means that the line connecting the focus and the vertex will have  slope -1/2.

Since you now know the slope (-1/2) and a point (1,-2) on the line connecting the focus with the vertex, you can write down the equation of that line.

The vertex itself will the the point of intersection between the line tangent at the vertex and the line connecting the focus and the vertex.
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Are you familiar with integration?
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