If α, β are the zeroes of the polynomial \(f(x)=x^2-p(x+1)-q\), then ( α+1)(β +1)=

Answer: \(( α+1)(β +1)=1-q\) Solution: Given: α, β are the zeroes of the polynomial \(f(x)=x^2-p(x+1)-q\). Step 1: Use the relationship between coefficients and roots of quadratic polynomial. \(f(x) =x^2-p(x+1)-q \) \(\Rightarrow f(x)=x^2-px-p-q \) \(\Rightarrow f(x)=x^2-px-(p+q) \) By comparing the polynomial \(f(x)=x^2-px-(p+q)\) with standard quadratic polynomial \(ax^2+bx+c\), we get \(a=1, b=-p\) and \(c=-(p+q)\). Sum of roots \(\left(\alpha+\beta\right)=-\frac{b}{a}=\) \(\Rightarrow \alpha+\beta=-\frac{-p}{1}=p\) Product of roots \((\alpha \cdot \beta)=\frac{c}{a}\) \(\Rightarrow \alpha\cdot\beta=\frac{-(p+q)}{1}=-p-q\) Step 2: Expand the expression \(\mathbf{(\alpha+1)(\beta+1)}\). \((\alpha+1)(\beta+1)=\alpha \cdot \beta+\alpha+\beta+1\) Substitute the value of \(\alpha \cdot \beta\) and (\alpha+\beta\), we get \((\alpha+1)(\beta+1)=p+(-p-q)+1\) \(\Rightarrow (\alpha+1)(\beta+1)=p-p-q+1\) \(\Rightarrow (\alpha+1)(\beta+1)=-q+1\) \(\Rightarrow (\alpha+1)(\beta+1)=1-q\) Hence, the value of \(\mathbf{(\alpha+1)(\beta+1)}\) is \(\mathbf{1-q}\).