Shift each partial product to the left to align the like terms when multiplying integers. An integer like 444 involves three different types of numbers: 4, 40 and 400. The position of a digit is part of its value. The multiplication algorithm first multiplies by the rightmost digit, then by the second digit from the right and so on. Each time, the top number is being multiplied by a number in a position that is 10 times as big. Shifting each partial product one place to the left aligns the like terms.
Collect the terms of the partial products of polynomial multiplication by aligning the terms that have the same exponents. Polynomial multiplication is the same as integer multiplication except that exponents are added and coefficients are multiplied. It is easy to line up the like terms because the exponents of the variable are what make two terms like terms, and these are easier to see than the position notation in integers. If both polynomials are arranged in standard form --- in decreasing order of exponents --- this usually involves shifting the partial products, as with integers; but if there are terms missing in the polynomials, the shifting may not be straightforward. To be safe, always align by the values of the exponents.
Align the partial products of the multiplication of complex numbers by looking at both the exponent of a term and whether it is real or imaginary. For example, iX is different from either i or X. Similarly, iX^2 is a different type from either i or X^2. When multiplying complex numbers, simple shifting almost never works. It will be necessary to align every partial product after the first one using two criteria: matching exponents and the presence or absence of an i --- an imaginary component.