The given equation is:

$x^2 + y^2 + 4z^2 + 12x = -35$

Step 1: Complete the square for the $x$-term.

The quadratic expression in $x$ is $x^2 + 12x$. To complete the square:

1. Take half of the coefficient of $x$, which is $12$, and divide by 2: $12/2 = 6$.
2. Square this result: $6^2 = 36$.

Now, we rewrite the $x^2 + 12x$ as: $x^2 + 12x = (x + 6)^2 - 36$

So the equation becomes: $(x + 6)^2 - 36 + y^2 + 4z^2 = -35$

Step 2: Simplify the equation.

Add 36 to both sides: $(x + 6)^2 + y^2 + 4z^2 = 1$

Step 3: Identify the surface.

This equation is of the form: $\frac{(x + 6)^2}{1} + \frac{y^2}{1} + \frac{z^2}{\frac{1}{4}} = 1$

This is the equation of an ellipsoid. The ellipsoid is centered at $(-6, 0, 0)$ and has semi-axes lengths of 1 in the $x$- and $y$-directions, and $1/2$ in the $z$-direction.

Summary:

The surface is an ellipsoid with center at $(-6, 0, 0)$, with semi-axes lengths 1 in the $x$- and $y$-directions, and $1/2$ in the $z$-direction.

Would you like more details or have any further questions?

Here are five related questions to deepen your understanding:

1. How would the equation change if the ellipsoid were centered at the origin?
2. What are the semi-axes lengths of the ellipsoid in general form?
3. How does the equation of an ellipsoid differ from that of a hyperboloid?
4. Can you transform this ellipsoid equation into cylindrical coordinates?
5. What is the geometric significance of completing the square in quadratic equations?

Tip: Completing the square is a useful technique for transforming quadratic equations into standard forms, making it easier to identify geometric properties.