A rectangle has one side that is 6 cm shorter than the other. The area of the rectangle will increas

Let the longer side be x cm. Since one side is 6 cm shorter, the other side is (x − 6) cm. The first area is x(x − 6) = x² − 6x.

Now add 2 cm to each side. The new sides are (x + 2) and (x − 6 + 2) = (x − 4). The new area is (x + 2)(x − 4) = x² − 2x − 8.

The area grew by 68 cm², so new area − old area = 68: (x² − 2x − 8) − (x² − 6x) = 68. The x² terms cancel, leaving 4x − 8 = 68, so 4x = 76 and x = 19.

That 19 cm is the longer side. The shorter side is 19 − 6 = 13 cm. Check: old area = 19 × 13 = 247; new area = 21 × 15 = 315; 315 − 247 = 68 ✓.