1. **State the problem:** We are given a $3 \times 3$ matrix: $$\begin{bmatrix} a+x & x & x \\ x & b+x & x \\ x & x & c+x \end{bmatrix}$$ and asked to verify or solve the determinant using characteristics of determinants, given the expression: $$\det = abc + x(ab + ac + bc)$$ 2. **Recall the determinant properties and formula:** The determinant of a $3 \times 3$ matrix with diagonal elements $a+x$, $b+x$, $c+x$ and off-diagonal elements $x$ can be expanded using the formula for determinants or by using properties such as multilinearity and symmetry. 3. **Calculate the determinant:** The determinant of the matrix is: $$\det = (a+x)(b+x)(c+x) + 2x^3 - x^2(a+b+c) - x^2(a+b+c)$$ But since the off-diagonal elements are all $x$, the determinant simplifies to: $$\det = (a+x)(b+x)(c+x) - x^2(a+b+c) + 2x^3$$ 4. **Expand $(a+x)(b+x)(c+x)$:** $$\begin{aligned} (a+x)(b+x)(c+x) &= abc + x(ab + ac + bc) + x^2(a + b + c) + x^3 \end{aligned}$$ 5. **Substitute back and simplify:** $$\begin{aligned} \det &= abc + x(ab + ac + bc) + x^2(a + b + c) + x^3 - x^2(a + b + c) + 2x^3 \\ &= abc + x(ab + ac + bc) + x^3 + 2x^3 \\ &= abc + x(ab + ac + bc) + 3x^3 \end{aligned}$$ 6. **Compare with given expression:** The given expression is: $$abc + x(ab + ac + bc)$$ Our calculation shows an extra $3x^3$ term. 7. **Conclusion:** The determinant of the matrix is: $$\boxed{abc + x(ab + ac + bc) + 3x^3}$$ which differs from the given expression by $3x^3$. This shows the importance of considering all terms in the determinant expansion, especially the cubic term from the off-diagonal elements. **Final answer:** $$\det = abc + x(ab + ac + bc) + 3x^3$$