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 $(x^2+y^2{)}^3=1444(x^4+2x^2{y}^2+y^4)-1444y^2(x^2+y^2)-1444y^4$.

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

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

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

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