$ \text{ If y -0, x = -7 or x = -4 the loop intersects the x-axis at x = -7 and at x=-4 } $

$ \text{ If S is the total length of the loop } $

$ S = 2 \int_{-7}{-4} \sqrt{1 + (\frac{dy}{dx}) ^2 } . dx $

$ 9y^2 = (x+7)(x+4)^2 $

$ 18 y \frac{dy}{dx} = (x+7) 2(x+4) + (x+4)^2 $

$ = (x+4)( 2x +14+x+4) $

$ = (x+4) (3x+18) = (x+4) 3(x+6) $

$ \frac{dy}{dx} = \frac{(x+4) (x+6)}{6y} $

$ 1+(\frac{dy}{dx})^2 = 1 + \frac{(x+4)^2(x+6)^2}{36y^2} $

$ 1 + \frac{(x+4)^2(x+6)^2}{4(x+7)(x+4)^2} $

$ = \frac{ 4x +28 +x^2 + 12x +36}{4(x+7)} $

$ \frac{(x+8)^2}{4 (x+7) } $

$ S = 2\int_{-1}^{-4} \frac{x+8}{2\sqrt s + 7 } $

$ \text{ Put } x+7 = t^2 \\ dx = 2tdt \\ x= -7, t = 0 \\ x= -4, t= \sqrt3 $

$ S = 2\int_0^(\sqrt 3) \frac{t^2 + 1} { 2t} 2t dt $

$ S = 2 \left [ \frac{t^3 }{3} + t ]_0^(\sqrt 3) \right ] = 4 \sqrt 3 $