Question

prove: an odd integer squared is odd.
(2n + 1)² = ?n² + n +
= (2n² + 2n) + 1
= odd

Explanation:

Step1: Expand the squared binomial

$$(2n+1)^2 = (2n)^2 + 2(2n)(1) + 1^2$$

Step2: Calculate each term

$$(2n+1)^2 = 4n^2 + 4n + 1$$

Step3: Factor out 2 from first two terms

$$(2n+1)^2 = 2(2n^2 + 2n) + 1$$

Step4: Verify odd number form

Since $2(2n^2 + 2n)$ is a multiple of 2 (even), adding 1 makes it odd.

Answer:

The filled-in steps are:
$(2n + 1)^2 = \boldsymbol{4}n^2 + \boldsymbol{4}n + \boldsymbol{1}$
$= \boldsymbol{2}(2n^2 + 2n) + 1$
This proves the square of an odd integer is odd.