Let's work through each problem step by step:

1. Find the center and radius of the circle and sketch its graph.

Given the equation:

$9x^2 + 36x + 9y^2 + 72y + 155 = 6$

First, we simplify this equation:

$9(x^2 + 4x) + 9(y^2 + 8y) + 155 = 6$

Subtract 155 from both sides:

$9(x^2 + 4x) + 9(y^2 + 8y) = -149$

Now, divide the entire equation by 9:

$x^2 + 4x + y^2 + 8y = -\frac{149}{9}$

Next, complete the square for both $x$ and $y$:

• For $x$, take half of 4 and square it: $(4/2)^2 = 4$.
• For $y$, take half of 8 and square it: $(8/2)^2 = 16$.

So, add and subtract these values inside the equation:

$(x^2 + 4x + 4) + (y^2 + 8y + 16) = -\frac{149}{9} + 4 + 16$

$(x + 2)^2 + (y + 4)^2 = \frac{-149 + 180}{9} = \frac{31}{9}$

Now, we have the equation of the circle in standard form:

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