Convert the following polar equation into a Cartesian equation. $r=-19 \sin (2 \theta)$

Step 1 ：First, we recall the conversion formulas from polar coordinates to Cartesian coordinates: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).

Step 2 ：Substitute \(r = -19 \sin(2\theta)\) into the conversion formulas, we get \(x = -19 \sin(2\theta) \cos(\theta)\) and \(y = -19 \sin(2\theta) \sin(\theta)\).

Step 3 ：We can simplify the above equations using the double-angle formulas: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

Step 4 ：Substitute \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\) into the equations, we get \(x = -38 \sin(\theta) \cos^2(\theta)\) and \(y = -38 \sin^2(\theta) \cos(\theta)\).

Step 5 ：We can further simplify the above equations using the Pythagorean identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\).

Step 6 ：Substitute \(\sin^2(\theta) = 1 - \cos^2(\theta)\) into the equation for \(y\), we get \(y = -38 (1 - \cos^2(\theta)) \cos(\theta)\).

Step 7 ：Simplify the equation for \(y\), we get \(y = -38\cos(\theta) + 38\cos^3(\theta)\).

Step 8 ：Now, we can eliminate \(\theta\) by squaring both equations and adding them together.

Step 9 ：We get \(x^2 + y^2 = 1444\cos^2(\theta) - 1444\cos^4(\theta)\).

Step 10 ：We can simplify the above equation using the identity: \(\cos^2(\theta) = 1 - \sin^2(\theta)\).

Step 11 ：Substitute \(\cos^2(\theta) = 1 - \sin^2(\theta)\) into the equation, we get \(x^2 + y^2 = 1444 - 1444\sin^2(\theta) - 1444\sin^4(\theta)\).

Step 12 ：We can further simplify the above equation using the identity: \(\sin^2(\theta) = y^2 / (x^2 + y^2)\).

Step 13 ：Substitute \(\sin^2(\theta) = y^2 / (x^2 + y^2)\) into the equation, we get \(x^2 + y^2 = 1444 - 1444(y^2 / (x^2 + y^2)) - 1444(y^4 / (x^2 + y^2)^2)\).

Step 14 ：Simplify the equation, we get \(x^2 + y^2 = 1444 - 1444y^2 / (x^2 + y^2) - 1444y^4 / (x^2 + y^2)^2\).

Step 15 ：Multiply the equation by \((x^2 + y^2)^2\), we get \((x^2 + y^2)^3 = 1444(x^2 + y^2)^2 - 1444y^2(x^2 + y^2) - 1444y^4\).

Step 16 ：Simplify the equation, we get \((x^2 + y^2)^3 = 1444(x^4 + 2x^2y^2 + y^4) - 1444y^2(x^2 + y^2) - 1444y^4\).

Step 17 ：Simplify the equation, we get \((x^2 + y^2)^3 = 1444x^4 + 2888x^2y^2 + 1444y^4 - 1444x^2y^2 - 1444y^4 - 1444y^4\).

Step 18 ：Simplify the equation, we get \((x^2 + y^2)^3 = 1444x^4 + 1444x^2y^2 - 1444y^4\).

Step 19 ：Divide the equation by 1444, we get \((x^2 + y^2)^3 / 1444 = x^4 + x^2y^2 - y^4\).

Step 20 ：Finally, we get the Cartesian equation: \(\boxed{(x^2 + y^2)^3 = 1444(x^4 + x^2y^2 - y^4)}\).