y=x²+12x+35=(x+5)(x+7) so the x-intercepts (when y=0) are -5 and -7.
y=x²+12x+36-36+35 is another way of writing the equation but then we can write:
y=(x+6)²-1. Since a square can never be negative the smallest value of (x+6)² is zero, when x=-6. And when x=-6, y=-1, the minimum value for y. So the vertex is (-6,-1), the minimum point on the parabola. Note that x=-6 is midway between the x-intercepts. These give us a clue in how to graph the parabola:
The vertical line x=-6 is the axis of symmetry. This acts like a mirror and reflects one half of the parabola exactly on to the other half.