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1 solution
3x^2 + 6x + 7= 3(x+1)^2 + 4, so sqrt (3x^2 + 6x + 7) >= 2.
5x^2 + 10x +14 = 5(x+1)^2 + 9, sqrt(5x^2 + 10x +14) >= 3.
The left hand side >=5.
4-2x-x^2 = 4 -(x+1)^2 +1 = 5-(x+1)^2 <= 5.
The right hand side <=5.
The solution exists only when (x+1)^2=0, so x=-1.