If x and y are the roots of the polynomial f(m)=m^2-p(m+ 1)-c, then findthe value of (x +1)(y +1)?

So you don't know what "root" means? You have a parabola there. It crosses the x axis in two places. If you plot the equation you have to put f(m) on the vertical axis and m on the horizontal axis. That is a confusion factor because we normally think of x and y being the axes, but this time they are not. So saying "x and y are the roots" is meaningless because the roots are expressed as values of m where f(m) = 0.

Let's play with this.
x^2 - p(x + 1) - c = 0

x^2 - px + (p - c) = 0 I write it this way to conform with our standard quadratic formula.

Negative b is the place to start,
Plus or minus will show you're smart,
The radical sits so gent-i-ly,
On b square minus 4ac
Now we're done ... except to say,
The whole thing sits on top of 2a

x = (p ± √(p^2 + 4p(p - c))/2

and:

y^2 - p(y + 1) - c = 0

I don't see where this is going anywhere.