determine the parameter m so that the vertices of the parabola, y=2x2+4x+m+1 lie on the line 5x+2y-1=0

First, we can use the vertex formula to find the x-coordinate of the vertex of the parabola$$ x=-\frac{b}{2a} $$$$ x=\frac{-4}{2\left(2\right)}=-1 $$Then we find the ordinate of the intersection of the vertex and the line. Using the equation of the line$$ 5x+2y-1=0 $$$$ 5\left(-1\right)+2y-1=0 $$$$ 2y=6 $$$$ y=3 $$So the vertex of the parabola which intersects the line is at (-1,3). Using the equation of the parabola, we find m.$$ y=2x^2+4x+m+1 $$$$ 3=2\left(-1\right)^2+4\left(-1\right)+m+1 $$$$ 3=2-4+m+1 $$$$ 4=m $$ Therefore, m = 4.