A Box In The Shape Of A Rectangular Prism Has A Volume Of 2500 Cm Cubed. The Box Is 4 Times As Wide As It Tall And 5 Cm Longer Than It Is Wide. What Are The Dimensions?

Unless you want to use the formula for the solution of a cubic, trial and error is "the right way."

Let t be the "tall" dimension.
t(4t)(4t+5) = 2500
16t³ +20t² -2500 = 0
4t³ + 5t² - 625 = 0

t=5 is a good first guess for a root. The other dimensions follow from that.

After finding the first root, the equation can be factored.
  = (t-5)(4t² + 25t + 125) = 0
The quadratic has no real roots, so t=5 is the only real solution to our cubic.