N2004 P1 Q5 [Modified]

The numbers \(x\) and \(y\) satisfy the equation \(4{{x}^{2}}+16xy+{{y}^{2}}+16x+14y+13=0\).

If a real value of \(x\) is substituted, the equation becomes a quadratic equation in \(y\). Given that two distinct real values of \(y\) may be found from this equation, show that

\(5{{x}^{2}}+8x+3>0\),

and hence find the set of possible values of \(x\).[5]

What can you deduce about the curve with equation

\(4{{x}^{2}}+16xy+{{y}^{2}}+16x+14y+13=0\)?[1]