The expression $x^2-y^2-z^2+2yz+x+y-z$ has:
$\textbf{(A)}\ \text{no linear factor with integer coefficients and integer exponents} \qquad \\ \textbf{(B)}\ \text{the factor }-x+y+z \qquad \\ \textbf{(C)}\ \text{the factor }x-y-z+1 \qquad \\ \textbf{(D)}\ \text{the factor }x+y-z+1 \qquad \\ \textbf{(E)}\ \text{the factor }x-y+z+1$
Factor the perfect square trinomial.
\[x^2 - (y-z)^2 + x + y - z\]
Factor the [difference of squares](https://artofproblemsolving.com/wiki/index.php/Difference_of_squares).
\[(x+y-z)(x-y+z) + x + y - z\]
Factor by grouping.
\[(x+y-z)(x-y+z+1)\]
The answer is $\boxed{\textbf{(E)}}$.