Answer
$a,b$
Work Step by Step
Step 1. Start from the original, expand the determinant using the last row:
$\begin{array}( \\|A|=-(1) \\ \\ \end{array}
\begin{vmatrix} a&x-a\\x&x \end{vmatrix}
\begin{array}( \\+(1) \\ \\ \end{array}
\begin{vmatrix} a&b\\x&x+b \end{vmatrix}$
Step 2. Complete the expansion:
$|A|=x(x-a)-ax+a(x+b)-bx=x^2-(a+b)x+ab=(x-a)(x-b)=0$
Step 3. We reach the solutions: $x=a,b$
