If α, β are the zeros of the polynomial f(x) = x2 − p(x + 1) − c

Since $\alpha$ and $\beta$ are the zeros of quadratic polynomial

$f(x)=x^{2}-p(x+1)-c$

$f(x)=x^{2}-p x-p-c$

$\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$

$=-\left(\frac{-p}{1}\right)$

$=p$

$\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$

$=\frac{-p-c}{1}$

$=-p-c$

We have

$0=(\alpha+1)(\beta+1)$

$0=\alpha \beta+(\alpha+\beta)+1$

$0=-c+1$

$c=1$

The value of $c$ is 1 .

Hence, the correct alternative is $(a)$