Literally Swapping Both Sides

Relevant wiki: Arithmetic Puzzles - Fill in the Blanks

Let us find the value that represents the red square first.

The last digit of $\color{#D61F06} \square \; \times \; 2$ is 8, so $\color{#D61F06}{\square}$ is either equal to 4 or 9. If $\color{#D61F06}{\square} = 9$ , then

$29 \; \times \; \color{#3D99F6}\square 2 = 288,$

dividing both sides of thr equation by 29 gives $\color{#3D99F6}\square 2 = \dfrac{288}{29} \approx 9.93$ . But since the left hand side of the equation is an integer but the right hand side of the equation is not an integer, which is impossible, so $\color{#D61F06}{\square} = 9$ cannot be true.

This leaves us with $\color{#D61F06}{\square} = 4$ only. We have

$24 \; \times \; \color{#3D99F6}\square 2 = 288.$

Dividing both sides of thr equation by 29 gives $\color{#3D99F6}\square 2 = \dfrac{288}{24} =12$ . So the digit that represents the blue square is 1. In other words, $\color{#D61F06}{\square} = 4$ and $\color{#3D99F6}{\square} = 1$ .

And the answer we're looking for is

$\color{#D61F06}{\square}2 \; \times \; 2 \color{#3D99F6}{\square} \; = \; 42 \; \times 21 \; = \; \boxed{882} \; .$

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Did you notice that the answer of 882 is literally the "reverse" of the number 288?

The unit digit of second term is 2 . The answers' unit digit is 8 . So you have to find a number such that, 2*x=8. Therefore red box is 4 . Now

$24\times(10x+2) = 288\text{, any two digit can be written as 10x+y. Here we keep blue box as x}\\ 10x+2=12\\ 10x=10\\ x=1$

Therefore two numbers are 24 , 12

So there answer would be

$42\times 21=882$

$2 \color{#D61F06}{4} × \color{#3D99F6}{1} 2 = 288$

$\color{#D61F06}{4}2 × 2 \color{#3D99F6}{1} = \boxed {882}$

I made a list of pairs that multiply to 288 (288,1) (144,2) (96,3) (72,4) (48,6) (36,8) (32,9) (24,12) (18,16)

only 24 and 12 satisfy the conditions.Imputing the numbers in the first equation and finding what each of the squares equal to and then putting that in the second equation which gives us 42x21 which is 882

First, we concluded that BLUE BOX cannot be greater than 1 since 21×22 greater than 228 . Hence we got the equation → 2×x×12=228 there fore x(red box) = 4 By putting the value we got 882. .. Q.E.D

It's ironical that the digits in the question as well as the answers are exactly reversed

2? * ?'2 = 288

Looking at the last digit of 288, I can say that ? * 2 = 8, therefore ? = 4

then it's just 288 / 24 = 12, so ?' = 1.

Plug these values into the question to find the answer