And beta, then we can write:

f(x) = (x-alpha)(x-beta)

Expanding this out, we get:

f(x) = x² - (alpha+beta)x + alpha*beta

Comparing this to the given expression for f(x), we see that:

p(1) = alpha+beta
c = alpha*beta

To find the value of p(2), we note that:

p(x) = f(x) + x

So:

p(2) = f(2) + 2

We can find f(2) by plugging in x=2 into the expression for f(x):

f(2) = 2² - p(1)*2 - c

Using the values we found earlier for p(1) and c, we have:

f(2) = 4 - (alpha+beta)*2 - alpha*beta

Now we need to express alpha+beta and alpha*beta in terms of p(2). To do this, we use the following identities:

(alpha+beta)² = alpha² + 2alpha*beta + beta² = (alpha+beta)² - 2alpha*beta
(alpha*beta)² = alpha²*beta² = (alpha*beta)²

Adding these two equations, we get:

(alpha+beta)² + (alpha*beta)² = (alpha+beta)² - 2alpha*beta + (alpha*beta)²

Simplifying, we have:

2(alpha*beta)² + 2alpha*beta = (alpha+beta)²

Using the values we found earlier for alpha+beta and alpha*beta, we have:

2c² + 2p(1)c = p(1)²

Substituting in the expression we found earlier for p(1), we have:

2c² + 2(p(2)-c)p(1) = p(1)²

Simplifying, we get:

2c² + 2p(2)p(1) - 2c*p(1) = p(1)²

Substituting in the expression we found earlier for c, we have:

2(alpha*beta)² + 2p(2)(alpha+beta) - 2(alpha*beta)(alpha+beta) = (alpha+beta)²

Simplifying, we get:

2(alpha+beta)(alpha*beta-p(2)) = 0

Since alpha and beta are distinct zeroes of f(x), we know that alpha+beta ≠ 0. Therefore, we must have:

alpha*beta - p(2) = 0

Solving for p(2), we get:

p(2) = alpha*beta = c

Therefore, we have:

p(2) = c = alpha*beta

as desired.