What does this mean?

P(x) / (x-7)  = x + 11/(x-7)      (given)  If you multiply both sides of this equation by (x-7) you will fid p(x)

p(x) =  x (x-7)   + 11

       = x^2 - 7x + 11      then put 7 in for 'x'  to find p(7) =  11

This is known as the "Remainder Theorem"

It says that   if we divide some polynomial P(x)  by   x - a, then the remainder will be equal to  P(a)

Cutting through the gobbledegook....let's look at an example

Suppose we divide  x^2 - x - 6     by    x - 3 

We have that

           x  + 2

x - 3 [  x ^2 - x - 6 ] 

           x^2  - 3x

           __________ 

                   2x - 6

                   2x - 6 

                  ______

                          0

Note that the remainder = 0

So....a = 3

And P(3)  =  (3)^2 - 3 - 6   =   0    ......the same as our remainder  !!!!! 

So..in your problem

a = 7   

p(x)   was divided by   x - a  and the remainder was 11

Which means that    p(7)   also =   11