Let $x_1,x_2 \in \mathbb{R}$ be the roots of the equation $x^2+px+q=0$ 
Let $x_1,x_2 \in \mathbb{R}$ be the roots of the equation
   $x^2+px+q=0$. Find $p$ and $q$ if it is known that $x_1+1$ and $x_2+1$
   are the roots of the equation $x^2-p^2x+pq=0$.

The roots of $x^2+px+q=0$ satisfy $$x_1+x_2=-\dfrac{b}{a}=-p,x_1x_2=\dfrac{c}{a}=q.$$ Also the roots of $x^2-p^2x+pq=0$ satisfy $$x_1+x_2=x_1+1+x_2+1=x_1+x_2+2=-\dfrac{b}{a}=p^2,x_1x_2=(x_1+1)(x_2+1)=\dfrac{c}{a}=pq.$$ Is it confusing I am using $x_1$ and $x_2$ for the roots of both equations? How can I improve what I've written? What to do next?

Another way :
Like If $p, q$ and $r$ are distinct roots of $x^3-x^2+x-2=0$, find the value of $p^3+q^3+r^3$.
let $x_k+1=y_k; k=1,2$
$\implies x_k=y_k-1$
As $x_k$ is a root of $$x^2+px+q=0$$
$$(y_k-1)^2+p(y_k-1)+q=0$$
