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The algorithm is as follows:
1. Find an equation of the straight line L1 through the given point perpendicular to the given straight line 2x + y = 0.
The line L1 is the altitude of the triangle.
Then find the intersection point of L1 and the other given line x-y+2 = 0.
You do it by mean solving the system of two linear equations in two unknown.
This intersection is the vertex V1 of the triangle.
All these operations are standard, and for the person who came with such a problem, it should not be difficult.
2. Do similar operations with the other line. I.E. :
Find an equation of the straight line L2 through the given point perpendicular to the given straight line x - y +2 = 0.
The line L2 is the second altitude of the triangle.
Then find the intersection point of L2 and the first given line 2x+y = 0.
You do it by mean solving the system of two linear equations in two unknown.
This intersection is the second vertex V2 of the triangle.
All these operations are standard, and for the person who came with such a problem, it should not be difficult.
3. Your last step is to find the equation of the straight line through the vertices V1 and V2.
The last line and last equation is/are what you are searching for.
Again, for the person who came with such a problem, all these operations and steps should be doable and feasible.
In any case, you got an algorithm from me.