You first have to put it in standard form, which is either
 
            (x - h)²      (y - k)²    
           ----------  + ---------- = 1 
               a²            b²
 
if it turns out that the larger number is under
the term on the left which contains x, and the 
graph will be an ellipse with a horizontal major
axis and a vertical minor axis, where
 
1. The center is (h, k) 
2. The length of the semi-major axis is " a ".
3. The ends of the major axis are the points (h-a,k) and (h+a,k)
4. The ends of the minor axis are the points (h,k-b) and (h,k+b)
5. Foci are (h-c,k), (h+c,k) where c is calculated from c² = a²-b²  
 
or
 
            (x - h)²      (y - k)²    
           ----------  + ---------- = 1 
               b²            a²
 
if it turns out that the larger number is under
the term on the left which contains y, and the 
graph will be an ellipse with a horizontal major
axis and a vertical minor axis, where and the 
graph will be an ellipse with a vertical major 
axis and a horizontal minor axis
 
where
 
1. The center is (h, k) 
2. The length of the semi-major axis is " a ".
3. The ends of the major axis are the points (h,k-a) and (h,k+a)
4. The ends of the minor axis are the points (h-b,k) and (h+b,k)
5. Foci are (h,k-a), (h,k+a) where c is calculated from c² = a²-b²  
 
We can't tell which it is unti we get it into standard form:
 
9x² + 16y² - 18x + 64y = 71
 
Rearrange getting x² term then the x term then y²and y terms
That is, swap the second and third terms:
 
9x² - 18x + 16y² + 64y = 71
 
Factor just the 9 out of the first two terms
on the left and also factor just the 16 out of 
the last two terms on the left: 
 
  9(x² - 2x) + 16(y² + 4y) = 71
 
Complete the square in the first parentheses:
 
The coefficient of x is -2
Multiply it by , getting -1
Square -1, getting +1.
Add +1 inside the first parentheses.
But the parentheses has a 9 in front.
So this amounts to adding 9·1 or 9 to
the left side, so we must add +9 to
the right side:
 
  9(x² - 2x + 1) + 16(y² + 4y) = 71 + 9
 
Complete the square in the second parentheses:
The coefficient of y is 4
Multiply it by , getting 2
Square 2, getting +4.
Add +4 inside the second parentheses.
But the parentheses has a 16 in front.
So this amounts to adding 16·4 or 64 to
the left side, so we must add +64 to
the right side:
 
  9(x² - 2x + 1) + 16(y² + 4y + 4) = 71 + 9 + 64
 
Factor both parentheses on the left and combine the 
numbers on the right
 
  9(x - 1)(x - 1) + 16(y + 2)(y + 2) = 144
 
              9(x - 1)² + 16(y + 2)² = 144
 
To get a 1 on the right side, divide every term
by 144
    
            9(x - 1)²   16(y + 2)²    144
           ---------  + ---------- = -----
               144         144        144
 
Cancel             
            1           1              1  
            9(x - 1)²   16(y + 2)²    144
           ---------  + ---------- = -----
               144         144        144
               16           9          1  
 
 
            (x - 1)²      (y + 2)²    
           ----------  + ---------- = 1 
               16            9        
 

Since 16 is greater than 9, and since a² is always the
larger of these in an ellipse, then a² is under the
term which contains x, we compare that to:
 
            (x - h)²      (y - k)²    
           ----------  + ---------- = 1 
               a²            b²
 
so it is an ellipse with a horizontal major axis and
a vertical minor axis. Comparing letters in this with
the numbers in the equation we got, we see that:
 
h = 1, k = -2, a² = 16, b² = 9 so a = 4 and b = 3
 
This means
1. The center is (h, k) = (1, -2)
2. The length of the semi-major axis is a = 4
3. The ends of the major axis (vertices) are the points (h-a,k), (h+a,k)
   or (1-4,-2) and (1+4,-2) or (-3,-2) and (5,-2)
4. The ends of the minor axis are the points (h,k-b), (h,k+b)
   or (1,-2-3), (1,-2+3) or (1,-5) and (1,1)
5. Foci are (h-c,k), (h+c,k) where c is gotten from c² = a²-b²  
   We calculate c:
   c² = a² - b²
   c² = 4² - 3²
   c² = 16 - 9
   c² = 7
   c = Ö7                                  _             _         
   So foci are (h-c,k) and (h+c,k), or (1-Ö7,-2) and (1+Ö7,-2), 
   or about (-1.6,-2) and (3.6,-2)
 
We draw the major and minor axes and plot the foci:   
 

 
Then we sketch in the ellipse:
 

 
The answers to your problem are:
 
center = (1, -2)
vertices = ends of major axis = (-3,-2) and (5,-2)
foci = (1-Ö7,-2) and (1+Ö7,-2), 
 
Edwin