Use the  factor  command for this:

factor(f);

                    A*sin(x)*theta(x)*(-m*omega^2+k)

By default, Maple calculates with 10 significant digits. Therefore, for a sufficiently large  x  of float type, we get ln(x+1) - ln(x) = 0. . If we increase the accuracy of the calculations, then the error disappears:

restart;
f := x->4*x*(ln(x+1)-ln(x)):
Digits := 50: 
evalf[10]~([seq(f(10.^n), n = 1 .. 10)]);

  [3.812407192, 3.980132341, 3.998001332, 3.999800013, 3.999980000, 3.999998000, 3.999999800, 3.999999980, 3.999999998, 4.000000000]

We can simply calculate this determinant and, equating it to  0 , we get a 4th degree equation for  w  with 2 parameters  d  and  k . Giving some values to these parameters, we find all values of  w . In the example below we find the real  d=2, k=1, w=sqrt(3)/3  for which  the determinant  is  0 :

Download det_new.mw

restart;
with(Student[Calculus1]):
soln := Roots(y-0.4646295e-3*tanh(y)+0.1839145082e-2*tanh(y)/(0.6000000000e-3*y^2-0.1840000000e-2), y);
map(t->t*I, soln);

p := 8*T(x, 7)*T(x, 2)+4*T(x, 5)*T(x, 1)+6*T(x, 3)*T(x, 3)+7*T(x, 1)*T(x, 4):
map(t->content(t)*`if`(type(primpart(t),`*`),convert(primpart(t),`+`),op(1,primpart(t))*op(2,primpart(t))), p);

                              8*T(x, 7)+8*T(x, 2)+4*T(x, 5)+11*T(x, 1)+12*T(x, 3)+7*T(x, 4)

Here is an another simple way (without select):

{seq(`if`(a>=5 and a<=15,a,NULL),a=A)};

move all terms to the left and take the numerator of the resulting fraction. In this case, the  expand  command is not required:

van_der_Waals := (p + a / V[m]^2) * (V[m] - b) = R * T:
numer((lhs-rhs)(van_der_Waals));
sort(%, V[m]);

restart;
Expr:=-a2/3 + a3 - (2*a4)/3 - (2*a5)/3:
Expr1:=-``(-Expr*3)/3; # The desired form
expand(Expr1); # The original form

Use functional notation:

Download CD_new.mw

2 errors in your code. In the EqBIS procedure, P, U, V must be Vectors, not lists. Also remove the space after the procedure name in your example (which Maple interprets as a multiplication sign). I also removed the unnecessary command RETURN:

EqBIS := proc(P, U, V)
local a, eq1, M1, t, PU, PV, bissec1;
a := (P - U)/LinearAlgebra:-Norm(P - U, 2) + (P - V)/LinearAlgebra:-Norm(P - V, 2);
M1 := P + a*t;
eq1 := op(eliminate({x = M1[1], y = M1[2]}, t));
op(eq1[2]); end proc;

EqBIS(<4, 5>, <11, 7/3>, <11, 5>);

restart;
F:=2*y*sin(beta*x)+6*z*cos(beta*x)+24*sin(beta*x)*cos(beta*x):
coeff(select(t->has(t,sin) and not has(t,cos),F), sin(beta*x));                              

In Maple gamma is Euler's constant, not a symbol. When you write  gamma(t) , this is a constant function, which equals  gamma  for any t .

restart;
simplify([3(t), gamma, gamma(t)]);
evalf(%);

                                            [3, gamma, gamma]
                               [3., 0.5772156649, 0.5772156649]
 

In the second example  gamma  is a symbol  (`γ`)

The procedure checks for a match in the rows of matrices  A  and  B  and returns a list of lists  L , in which the 1st element is the matching row, and the 2nd element is a list of two numbers (how many times this row occurs in each of the matrices).
 

Download CheckRowsUniquie.mw

Just use the  RelabelVertices  command as in the example below:

restart; 
with(GraphTheory): 
G := Graph({{1, 2}, {1, 3}, {1, 4}}); 
V := Vertices(G); 
V1 := subs([1, 2]=~[a, b], V); 
H := RelabelVertices(G, V1); 
DrawGraph(H);

Just specify the ranges for the variables  x  and  y  and the error will disappear:

restart;
plots:-implicitplot(x-y-Pi, x=-1..5, y=-4..1);