To find the distance between points R and S, we need to use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by:

√[(x2 - x1)^2 + (y2 - y1)^2]

Here, R is the point (a, b) and S is the point (a - b, -a - b). Let's substitute these values into the distance formula and simplify the expression step by step.

**Step 1: Substitute the values into the distance formula.**

The coordinates of R are (a, b) and the coordinates of S are (a - b, -a - b). Plugging these values into the distance formula, we get:

√[(a - b - a)^2 + (-a - b - b)^2]

**Step 2: Simplify the expression.**

Expanding the square terms, we have:

√[(-b)^2 + (-2a - 2b)^2]

Simplifying further, we get:

√[b^2 + (2a + 2b)^2]

**Step 3: Expand the square terms.**

Expanding the square terms, we have:

√[b^2 + (2a)^2 + (2b)^2 + 2(2a)(2b)]

Simplifying further, we get:

√[b^2 + 4a^2 + 4b^2 + 8ab]

**Step 4: Combine like terms.**

Combining like terms, we have:

√[5b^2 + 4a^2 + 8ab]

Therefore, the distance between points R and S is √[5b^2 + 4a^2 + 8ab].

In conclusion, the distance between points R and S is given by the expression √[5b^2 + 4a^2 + 8ab].