Verify the Factorization: Is x^2+2x+1 = (x+2)(x-1)?

Let's try to factor using quick trinomial factoring the given expression:

$x^2+2x+1$

We'll look for a pair of numbers whose product is the free term in the expression, and their sum is the coefficient of the first power term in the expression, meaning two numbers $m,\hspace{2pt}n$ that satisfy:

$m\cdot n=1\\ m+n=2\\$From the first requirement mentioned, namely - from the multiplication, we should note that the product of the numbers we're looking for needs to yield a positive result, therefore we can conclude that both numbers have the same signs, according to multiplication rules, and now we'll remember that the possible factors of 1 are 1 and -1, fulfilling the second requirement mentioned, along with the fact that the signs of the numbers we're looking for are equal to each other will lead to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases} m=1\\ n=1\end{cases}$

and therefore we'll factor the given expression to:

$x^2+2x+1 \\ \downarrow\\ (x+1)(x+1)$

and therefore clearly the factorization suggested in the problem is incorrect.

Therefore - the correct answer is answer B.

Note:

The given question could also be solved by expanding the parentheses in the suggested expression:

$(x+2)(x-1)$ (using the expanded distributive property), and checking if we indeed get the given expression:

$x^2+2x+1$, however it is of course preferable to try to factor the given expression - for practice purposes.