The area of a triangle is 20 cm^2, and the altitude is 4 cm greater than the base. find the length of the base, to the nearest millimeter.

Area of any triangle = (1/2) x (base) x (height)

You said ...

Area = 20 cm²
Height = (b + 4)

so we can write the area formula like this,
with everything in centimeters:

                                                          20 = (1/2) x (b) x (b + 4)

Eliminate the parentheses on the right:  20 = (1/2) (b² + 4b)

Multiply each side by  2 :                     40 = b² + 4b

Subtract  40  from each side:              b² + 4b - 40 = 0

Use the quadratic formula to solve this equation,
and you find that

             b = -2 plus or minus 6.633

             b = 4.633
             b = -8.633

The meaningful solution is    base = 4.633 cm

Check:

Area = (1/2) (b) (b+4) =

       (1/2) (4.633) (8.633) =

                (1/2) (40 cm²) = 20 cm²                yay !