From the first equation, y=x-4, so we substitute this value of y in the second equation:
x^2+5y=4 becomes x^2+5(x-4)=4; x^2+5x-20=4; x^2+5x-24=0=(x+8)(x-3)=0.
Therefore, x=-8 or 3, making y=x-4=-12 or -1. So the solutions are (x,y)=(-8,-12) or (3,-1).
The problem can be solved graphically, too. Draw the graphs of y=x-4 (straight line) and 5y=4-x^2 (parabola) and you'll find the line cuts the parabola at (-8,-12) and (3,-1).
The line y=x-4 graph is drawn by marking -4 on the y axis and 4 on the x axis (the intercepts) and drawing a line through these points, extending it beyond these points.
The parabola is an inverted U shape. It intersects the x axis at 2 and -2 and its vertex (maximum point) is at y=4/5 the point (0,4/5). You should be able to see how the line intersects the parabola at two points.