Given that $(x+1)$ and $(x-1)$ are factors of the polynomial $2x^3 - ax^2 + bx - 5$, we can use the fact that if $x - c$ is a factor of a polynomial $P(x)$, then $P(c) = 0$.

Step 1: Apply the Factor Theorem

Since $(x+1)$ is a factor, substituting $x = -1$ into the polynomial should give us 0:

$P(-1) = 2(-1)^3 - a(-1)^2 + b(-1) - 5 = 0$

Simplify this:

$-2 - a - b - 5 = 0$

$- a - b = 7 \implies a + b = -7 \quad (1)$

Similarly, since $(x-1)$ is a factor, substituting $x = 1$ should also give us 0:

$P(1) = 2(1)^3 - a(1)^2 + b(1) - 5 = 0$

Simplify this:

$2 - a + b - 5 = 0$

$-a + b = 3 \quad (2)$

Step 2: Solve the System of Equations

We now have a system of two equations:

1. $a + b = -7$
2. $-a + b = 3$

Add these two equations together:

$(a + b) + (-a + b) = -7 + 3$

$2b = -4 \implies b = -2$

Substitute $b = -2$ into equation (1):

$a + (-2) = -7$

$a = -5$

Step 3: Conclusion

The values of $a$ and $b$ are:

$\boxed{a = -5, \; b = -2}$

Would you like to know more about polynomial factorization or any other topic? Here are some related questions:

1. How to factor polynomials completely?
2. What is the Factor Theorem, and how is it used?
3. How do you solve a system of linear equations?
4. What are some practical applications of polynomials in real life?
5. How to find the roots of a cubic polynomial?

Tip: Always double-check your substitution and simplification steps to avoid errors when solving equations!