41. $$ x^{2}+5 y^{2}-8 x-30 y-37=0 $$ 42. $$ 3 x^{2}+y^{2}+18 x-2 y-8=0 $$

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41. Start with 
$$
x^{2}+5y^{2}-8x-30y-37=0.
$$
Group the $$x$$ and $$y$$ terms:
$$
(x^{2}-8x)+5(y^{2}-6y)-37=0.
$$
Complete the square for the $$x$$-terms. For $$x^2-8x$$, take half of $$-8$$ which is $$-4$$ and square it to get $$16$$. Add and subtract $$16$$:
$$
(x^{2}-8x+16-16)+5(y^{2}-6y)-37=0.
$$
Now complete the square for the $$y$$-terms. Factor the $$5$$ first:
$$
5\Bigl[y^{2}-6y\Bigr]=5\Bigl[(y^{2}-6y+9-9)\Bigr],
$$
since half of $$-6$$ is $$-3$$ and $$(-3)^2=9$$. This gives:
$$
(x^{2}-8x+16)+5\Bigl[(y-3)^2-9\Bigr]-16-37=0.
$$
Rewrite the squares:
$$
(x-4)^2+5(y-3)^2 -16-45-37=0.
$$
Combine the constants:
$$
(x-4)^2+5(y-3)^2 -98=0.
$$
Thus, the equation in completed square form is:
$$
(x-4)^2+5(y-3)^2=98.
$$
This is the standard form representing an ellipse.

42. Begin with
$$
3x^{2}+y^{2}+18x-2y-8=0.
$$
Group the $$x$$ and $$y$$ terms:
$$
3x^{2}+18x + y^{2}-2y-8=0.
$$
For the $$x$$-terms, factor out $$3$$:
$$
3\Bigl(x^{2}+6x\Bigr)+y^{2}-2y-8=0.
$$
Complete the square for $$x^{2}+6x$$. Half of $$6$$ is $$3$$ and $$3^{2}=9$$:
$$
3\Bigl[(x^{2}+6x+9)-9\Bigr]+y^{2}-2y-8=0,
$$
which becomes
$$
3(x+3)^2-27+y^{2}-2y-8=0.
$$
Now complete the square for the $$y$$-terms. For $$y^{2}-2y$$, half of $$-2$$ is $$-1$$ and $$(-1)^2=1$$:
$$
3(x+3)^2-27+[(y^2-2y+1)-1]-8=0.
$$
Write the squared term:
$$
3(x+3)^2-27+(y-1)^2-1-8=0.
$$
Combine the constants:
$$
3(x+3)^2+(y-1)^2-36=0.
$$
Add $$36$$ to both sides:
$$
3(x+3)^2+(y-1)^2=36.
$$
To put this in standard form, divide the entire equation by $$36$$:
$$
\frac{3(x+3)^2}{36}+\frac{(y-1)^2}{36}=1,
$$
which simplifies to
$$
\frac{(x+3)^2}{12}+\frac{(y-1)^2}{36}=1.
$$
This is the standard form of an ellipse.
41. The equation $$ x^{2}+5y^{2}-8x-30y-37=0 $$ can be rewritten in standard form as $$ (x-4)^2 + 5(y-3)^2 = 98 $$, representing an ellipse. 42. The equation $$ 3x^{2}+y^{2}+18x-2y-8=0 $$ simplifies to $$ \frac{(x+3)^2}{12} + \frac{(y-1)^2}{36} = 1 $$, which is also an ellipse.

41. \( x^{2}+5 y^{2}-8 x-30 y-37=0 \) 42. \( 3 x^{2}+y^{2}+18 x-2 y-8=0 \)

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