Question 709326
As briefly as possible,

L=w+10
d+w=50  (but the description from posted question is not specific enough)


Area of the sides: {{{2d(L+w)}}}
Area of top and base: {{{2Lw}}}


For area of the sides, the process was, 2Ld plus 2dw or 2dL+2dw=2d(L+w).


"the combined area of the top and the base is 400 in^2 less than the total area of the four sides, ", must mean:
{{{2Lw=2d(L+w)-400}}}.
First simplification is to divide by 2.
{{{highlight(Lw=d(L+w)-200)}}}


Used next the facts, {{{L=w+10}}} and {{{d+w=50}}}, substituted and several algebra steps, solving all into terms of L and constants, ...
...
{{{highlight(L^2-40L+260=0)}}}


Quadratic formula solution needed to be used.  The simple radical form found for L was {{{20+2*sqrt(35)}}} or {{{20-2*sqrt(35)}}}.  The minus-form turned to be a non-useful result.


Length then was {{{highlight(20+2*sqrt(35)=8.168)}}} (approximate).
Using the other two facts, {{{highlight(w=21.832)}}}, {{{highlight(d=28.168)}}}.