1. Solve the equations $15x - y = 82$ and $6x - 5y = -4$

Multiply the first equation by 5 and the second by 1 and then subtract the second from the first:

$75x - 5y = 410$

$6x - 5y = -4$

Subtracting these gives $69x = 414$ which leads to $x = 6$

Substitute $x = 6$ into the first equation to get $y = 8$

So, the first vertex is $(6,8)$.

1. Solve the equations $15x - y = 82$ and $9x + 4y = 17$

Multiply the first equation by 4 and the second by 1 and then add:

$60x - 4y = 328$

$9x + 4y = 17$

Adding these gives $69x = 345$ which leads to $x = 5$

Substitute $x = 5$ into the second equation to get $y = -7$

So, the second vertex is $(5,-7)$.

1. Solve the equations $6x - 5y = -4$ and $9x + 4y = 17$

Multiply the first equation by 4 and the second by 5 and then add:

$24x - 20y = -16$

$45x + 20y = 85$

Adding these gives $69x = 69$ which leads to $x = 1$

Substitute $x = 1$ into the first equation to get $y = 2$

So, the third vertex is $(1,2)$.

So, the vertices of the triangle are $(6,8)$, $(5,-7)$, and $(1,2)$.

Now that we have the vertices of the triangle, we can find the centroid. The centroid $(\alpha, \beta)$ of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by :

$$ \alpha=\frac{x_1+x_2+x_3}{3}$$,

$$ \beta=\frac{y_1+y_2+y_3}{3} $$

Substituting the coordinates of the vertices into these equations, we get:

$$ \alpha=\frac{6+5+1}{3}=4 $$,

$$ \beta=\frac{8-7+2}{3}=1 $$

Then, we find $\alpha+2\beta$ and $2\alpha-\beta$:

$$ \alpha+2\beta=4+2(1)=6 \ 2\alpha-\beta=2(4)-1=7 $$

So, $\alpha+2\beta=6$ and $2\alpha-\beta=7$ are the roots of the quadratic equation.

We can write the quadratic equation as

$$x^2-(\alpha+2\beta+2\alpha-\beta)x+(\alpha+2\beta)(2\alpha-\beta)=0$$

Substituting $\alpha=4$, $\beta=1$ gives us:

$$x^2-13x+42=0$$