Ethiopian Multiplication

Python 2, 203 202 187 133 bytes

a,b=input()
s=0
while a:print'%3s%9s'%(a,'[ %%dd] '[a%2::2]%b);s+=[0,b][a%2];a/=2;b*=2
print'%10s==\n%11s'%(''.rjust(len(`s`),'='),s)

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If I can use * for string multiplication ('='*R) and as a 'selector' (b*(a%2) instead of [0,b][a%2]), I get:

118 bytes

a,b=input()
s=0
while a:print'%3s%9s'%(a,'[ %%dd] '[a%2::2]%b);s+=a%2*b;a/=2;b*=2
print'%10s==\n%11s'%('='*len(`s`),s)

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Explanation:

a,b=input()                   #Get input
L=len(`a`)                    #Get length of first number for adjusting text
l=[]                          #Output list
s=0                           #Sum
while a:
 B=['[%d]',' %d '][a%2]%b     #B is either '[b]' or ' b ' depending on if a is odd/even
 l+=[(`a`,B)]                 #Add a,B to output list
 s+=[0,b][a%2]                #Add b to sum if a is odd
 a/=2;                        #Halve a
 b*=2;                        #Double b
R=len(B)                      #Length of last B for adjusting output
l+=[('',''.rjust(R,'='))]     #Add double line ==== to output list
l+=[('','%d '%s)]             #Add sum to output list
for x,y in l:
 print x.rjust(L),y.rjust(R)  #Print adjusted numbers

Charcoal, 91 bytes

≔⟦⟧τ≔⁰σＮθＮηＷθ«⊞τ⪫  Ｉθ⊞υ⪫⎇﹪θ²  ¦[]Ｉη≔⁺σ∧﹪θ²ησ≔÷θ²θ≔⁺ηηη»⊞υ…=⁺²ＬＩσ⊞υ⪫  Ｉσ←Ｅ⮌τ⮌ιＭ⌈ＥυＬιＬυ←Ｅ⮌υ⮌ι

Try it online! Link is to verbose version of code. Explanation:

≔⟦⟧τ≔⁰σ

Sets t to the empty list and s to 0. (u already defaults to the empty list.)

ＮθＮη

Inputs the two numbers.

Ｗθ«

Repeats while q is nonzero.

   ⊞τ⪫  Ｉθ

Wrap q in padding and append it to the list t.

   ⊞υ⪫⎇﹪θ²  ¦[]Ｉη

Wrap h either in padding or [] depending on whether q is odd, and append it to the list u.

   ≔⁺σ∧﹪θ²ησ

Add h to s if q is odd.

   ≔÷θ²θ

Integer divide q by 2.

   ≔⁺ηηη»

Add h to itself.

⊞υ…=⁺²ＬＩσ

Append a suitable string of = signs to the list u.

⊞υ⪫  Ｉσ

Append the padded sum s to the list u.

←Ｅ⮌τ⮌ι

Rotate the list t by 180° and print it up-side down, thus right-justifying it.

Ｍ⌈ＥυＬιＬυ←Ｅ⮌υ⮌ι

Move the cursor so that when u is right-justified its top left corner lines up with the top-right corner we just reached, and print u right-justified.

Java (OpenJDK 8), 353 316 267 214 210 bytes

(a,b)->{int g=0;for(;a>0;g+=a%2*b,a/=2,b*=2)System.out.printf("%1$8d%2$10s\n",a,a%2<1?"["+b+"]":b+" ");System.out.printf("%1$19s%2$18s","".valueOf(new char[(int)Math.log10(g)+3]).replace("\0","=")+"\n",g+" ");}

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