Question 35675
Start with the formula for the area of a trapezoid:
{{{A = ((b1 + b2)/2)h}}}
In your problem:
A = 48 sq.cm.
b1 = x+6
b2 = x
h = x+1

{{{48 = (((x+6)+(x))/2)(x+1)}}} Simplify.
{{{48 = ((2x+6)/2)(x+1)}}} Factor a 2 in the top.
{{{48 = (2(x+3)/2)(x+1)}}} Cancel the 2s.
{{{48 = (x+3)(x+1)}}} Perform the indicated multiplication.
{{{48 = x^2+4x+3}}} Subtract 48 from both sides of the equation.
{{{x^2+4x-45 = 0}}} Solve for x by factoring.
{{{(x-5)(x+9) = 0}}} Apply the zero products principle.
{{{x-5 = 0}}} and/or {{{x+9 = 0}}}
If {{{x-5 = 0}}} then {{{x = 5}}} This is an acceptable solution as it is positive.
If {{{x+9 = 0}}} then {{{x = -9}}} This is not meaningful as the length of the base has to be a positive value.

Answer: 
x = 5 cm.