@Thomas Richard  Thank you very much! The third item on the list "Questions" helped.

I have the same problem in 32 bit Maple 2017.3 on Windows 10. The problem appeared about 2.5 months ago. Since then, I have reinstalled Maple and even been able to update it to Maple 2017.3, but the problem has remained (the standard GUI hangs on boot).

@pgl771  This is even simplier:

plot3d(y^2+x, x = 0 .. 2, y = x-2 .. x, orientation = [220, 75]);

@Les  For plotting in geom3d  package the  draw  command should be used:

draw(pp);

@Carl Love  The Iterator package was introduced in Maple 2016, therefore, unfortunately, OP will not be able to check the operation of your procedure.

@mapleatha  I was able to get only this option (with lambda[1]  in front of I):

restart;
with(InertForm):
Display(`%+`(A,`.`(lambda[1], convert(I,symbol))))=<a,b; c,d>;

@nunavutpanther   You can write this shorter:

L1:=[y,x*y];
L2:=[x^2,x*y];
{L1[], L2[]};

@alfarunner  Do not forget  restart  before running the code. For me in Maple 2015 - 2017 everything works properly:

restart;
z:=(-1/(4*(-u+L)))*(4*p^2*b^2*d-3*g*c*p^2);
u:=2:  L:=1:  c:=0.6: 
constr:={ 1-(2*d)/(p*(sqrt(g)+1))<=b, b<=1, 0<=d, d<=L, c<=p, p<=u, 1<=g, g<=4};  
Optimization:-Maximize(z, constr);

I tried to apply your procedures (in Maple 2017.3 and 2016.2) to the same example and got a strange error:

restart;
ranges:=proc(simpledom::list(relation), X::list(name))
local rez:=NULL, r,z,k,r1,r2;
if nops(simpledom)<>2*nops(X) then error "Domain not simple!" fi;
for k to nops(X) do    r1,r2:=simpledom[2*k-1..2*k][]; z:=X[k];
  if   rhs(r1)=z and lhs(r2)=z then rez:=z=lhs(r1)..rhs(r2),rez; #a<z,z<b
  elif lhs(r1)=z and rhs(r2)=z then rez:=z=lhs(r2)..rhs(r1),rez  #z<b,a<z
  else error "Strange order in a simple domain" fi
od;
rez
end proc:

MultiIntPoly:=proc(f, rels::list(relation(ratpoly)), X::list(name))
local r,rez,sol,irr,wirr, rels1, w;
irr:=[indets(rels,{function,realcons^realcons})[]];
wirr:=[seq(w[i],i=1..nops(irr))];
rels1:=eval(rels, irr=~wirr);
sol:=SolveTools:-SemiAlgebraic(rels1,X,parameters=wirr):
sol:=remove(hastype, eval(sol,wirr=~irr), `=`); 
add(Int(f,ranges(r,X)),r=sol)
end proc:

R := [ (x-4)^2+y^2 <= 25, x^2+(y-3)^2 >= 9, (x+sqrt(7))^2+y^2 >= 16 ];
MultiIntPoly(1, R, [x,y]): 
radnormal(simplify(value(%)));

@mapleatha  In this case use  Standard GUI  in which there are no any parentheses around the exponent.

In Classic interface, you can do so:

expa := 2: 
exp(cat(expa,` t`));

Edit.

@vv  For better perception, I did everything on one plot, eliminated some repetitions in the code and added the  fieldstrength = log   option that equalizes the lengths of the arrows:

restart;
with(plots): with(plottools):
poly:=x>=0, y<=1, y>=-1, y<=4-4*x, y>=4*x-4:
F,G:=piecewise~(`and`(poly), [x^2+y^2,3*sin(x)-y^2], undefined)[]:
display(polygon([[0, -1], [3/4, -1], [1, 0], [3/4, 1], [0, 1]], style = line, thickness = 2), fieldplot([F,G], x =0 .. 1, y = -1 .. 1, arrows = SLIM, color = x, fieldstrength = log));

@nunavutpanther If  X  is a list then  add(X)  only works in the latest versions of Maple. Replace   add(X)  with  `+`(X[ ])

@digerdiga 

expression:=(a*b^2*c*d+a^2*b*c*e)*x^2 + (a*b+a^2*e)*x:
map(factor, expression);
 

@Joe Riel   From the condition of the problem  "...carpet  covers exactly 24 plates and at the same time each vertical and each horizontal row of plates in the living room contains 
exactly 4 plates covered with carpet..." it follows that the room size is exactly  6 by 6

In the general case, we also have 5 roots. So what's the problem?

f:=1/4*(-1/4+alpha*(-1+b)*e^2+((1-b)*alpha+b/4)*e)^2/(-1+e)^2/(b*e-1)/alpha/e/(-1+b):
diff(f, e):
solve(%, e):
nops([%]);
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