Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
(A) write down the equation of the line of symmetry,
(B) write down the minimum \(y\)-value on the curve.