We are tasked with converting the general form equation of a hyperbola:

$8x^2 - 5y^2 + 16x - 32 = 0$

to its standard form. Follow these steps:

Step 1: Group the $x$ and $y$ terms.

We start by grouping the $x$-terms and $y$-terms:

$(8x^2 + 16x) - 5y^2 = 32$

Step 2: Factor out the coefficient of $x^2$.

We factor out $8$ from the $x$-terms:

$8(x^2 + 2x) - 5y^2 = 32$

Step 3: Complete the square for the $x$-terms.

To complete the square for $x^2 + 2x$, we take half of $2$, which is $1$, and square it to get $1$. So, we add and subtract $1$ inside the parentheses:

$8(x^2 + 2x + 1 - 1) - 5y^2 = 32$

Simplifying:

$8((x + 1)^2 - 1) - 5y^2 = 32$

Step 4: Distribute the constants.

Now, distribute the $8$:

$8(x + 1)^2 - 8 - 5y^2 = 32$

Simplify further:

$8(x + 1)^2 - 5y^2 = 40$

Step 5: Divide by the constant on the right-hand side.

To convert this into standard form, we divide everything by $40$:

$\frac{8(x + 1)^2}{40} - \frac{5y^2}{40} = 1$

Simplifying the fractions:

$\frac{(x + 1)^2}{5} - \frac{y^2}{8} = 1$

Final Standard Form:

$\frac{(x + 1)^2}{5} - \frac{y^2}{8} = 1$

This is the standard form of the given hyperbola.

Would you like more details on any specific step, or do you have any questions?

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Here are 5 related questions:

1. How do you determine the center of a hyperbola from its standard form?
2. What are the key differences between the equations of ellipses and hyperbolas?
3. How do you calculate the foci of a hyperbola given its standard form?
4. Can the orientation of a hyperbola change based on its equation? How?
5. How is the asymptote of a hyperbola derived from its standard form?

Tip: For hyperbolas, the terms involving squares have opposite signs (one positive, one negative), which distinguishes them from ellipses.