In a binomial expansion, what is the coefficient of x² in the expansion of (x + 2)³?

To find the coefficient of ( x^2 ) in the expansion of ( (x + 2)^3 ), we can utilize the binomial theorem, which states that the expansion of ( (a + b)^n ) can be expressed as:

[

\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

]

In this case, ( a = x ), ( b = 2 ), and ( n = 3 ). We are interested in the term where ( x ) is raised to the power of 2. In the binomial expansion, the general term is given by:

[

\binom{3}{k} x^{3-k} (2)^k

]

To find the coefficient of ( x^2 ), we need to set ( 3 - k = 2 ), which implies ( k = 1 ). Now we can plug this value into the formula:

[

\text{Term when } k = 1:\quad \binom{3}{1} x^{2} (2)^1

]

Calculating the coefficient:

1. ( \bin