One important thing to remember in these problems is that 
we must shift vertically UP or DOWN last.

We start with

y = x2


Then we stretch it vertically by a factor of 5 by multiplying
the right side by 5 and we get

y = 5x2


Then we shift it horizontally to the left four units by 
replacing x by (x+4) in the right side and we get:

y = 5(x+4)2


Then finally we shift it vertically up by 1 unit by adding 1 
to the right side.:

y = 5(x+4)2+1


And that is the graph of

f(x) = 5(x+4)2+1

Edwin