a)

 x² + 10x - 6y + 7 = 0

Since there is an x² term, isolate the terms in x on the left
side of the equation:

 x² + 10x = 6y - 7 

If x² had not had a coefficient of 1 then we woulfd
have to divide through by it.  But since x² has a 1
coefficient this is not necessary]

 x² + 10x = 6y - 7 

We complete the square:

1. Multiply the coefficient of x which is +10 by  getting +5
2. Square this, getting (+5)² = +25
3. Add this to both sides of the equation:

 x² + 10x + 25 = 6y - 7 + 25

The left side factors as (x+5)(x+5) or (x+5)²
Combine like terms on the right side of the equation.

(x + 5)² = 6y + 18

Finally factor out the coefficient of y on the right

(x + 5)² = 6(y + 3)

That's the answer, but later you'll have to compare it to

(x - h)² = a(y - k)

And the vertex is (h,k) = (-5,-3) and since x is squared and not y,
the parabola has a vertical axis of symmetry and since a = 6 and
is positive the parabola opens upward.

------------------------------
 
(b) 

 y² + 8x - 2y = 15

Since there is a y² term, isolate the terms in y on the left
side of the equation:

 y² - 2y = -8x + 15 

If y² had not had a coefficient of 1 then we woulfd
have to divide through by it.  But since y² has a 1
coefficient this is not necessary]

 y² - 2y = -8x + 15 

We complete the square:

1. Multiply the coefficient of y which is -2 by  getting -1
2. Square this, getting (-1)² = +1
3. Add this to both sides of the equation:

 y² - 2y + 1 = -8x + 15 + 1

The left sides factors as (y-1)(y-1) or (y-1)²
Combine like terms on the right side of the equation.

(y - 1)² = -8x + 16

Finally factor out the coefficient of x on the right

(y - 1)² = -8(x - 2)

That's the answer, but later you'll have to compare it to

(y - k)² = a(x - h)

And the vertex is (h,k) = (2,1) and since y is squared and not x,
the parabola has a horizontal axis of symmetry and since a = -8 and
is negative the parabola opens to the left.

Edwin