What should be added to $ 3{x^3} - 2{x^2} + 5x + 1 $ so that the sum becomes $ {x^3} - 2{x^2} + 4x - 1 $ ?

Hint: The expression given should be compared to the expression which is required. Based on the comparison the suitable term should be chosen.

Complete step-by-step answer:
The given expression is
 $ 3{x^3} - 2{x^2} + 5x + 1 $
The first term of the given expression is $ 3{x^3} $ and the first term of the required expression is $ {x^3} $ . So the term $ \left( { - 2{x^3}} \right) $ should be added to $ 3{x^3} $ .
The second term of the given expression is $ - 2{x^2} $ and second term of the required expression is $ - 2{x^2} $ . Both the terms are the same .So, no addition of the terms are required.
The third term of the given expression is $ 5x $ and the third term of the required expression is $ 4x $ . So the term (-x) should be added to $ 5x $ .
The fourth term of the given expression is $ 1 $ and the fourth term of the required expression is $ - 1 $ . So the term $ \left( { - 2} \right) $ should be added to $ 1 $ .
Hence, the combined term which should be added to $ 3{x^3} - 2{x^2} + 5x + 1 $ is $ - 2{x^3} - x - 2 $ to make $ {x^3} - 2{x^2} + 4x - 1 $

Note: It can solved by another method
Let’s suppose the term $ B $ that is to be added to $ 3{x^3} - 2{x^2} + 5x + 1 $ , to make the expression equal to
 $ {x^3} - 2{x^2} + 4x - 1 $
According to the condition of the question,
 $ \left( {3{x^3} - 2{x^2} + 5x + 1} \right) + B = {x^3} - 2{x^2} + 4x - 1 $
Solving the above expression for $ B $ ,
\[
  \left( {3{x^3} - 2{x^2} + 5x + 1} \right) + B = \left( {{x^3} - 2{x^2} + 4x - 1} \right) \\
  B = \left( {{x^3} - 2{x^2} + 4x - 1} \right) - \left( {3{x^3} - 2{x^2} + 5x + 1} \right) \\
  B = {x^3} - 3{x^3} - 2{x^2} + 2{x^2} + 4x - 5x - 1 - 1 \\
  B = - 2{x^3} - x - 2 \\
 \]
Hence, the expression to be added is $ - 2{x^3} - x - 2 $ .