Final answer:
To make the expression 4×square + ⅔ a perfect square, the term to be inserted in the expression is 1. Therefore, the completed expression will be (2×square + 1)^2.
Step-by-step explanation:
The term to insert in the expression 4×square + ⅔ to make it a perfect square. To turn this expression into a perfect square, we need to find a value that, when squared, produces the constant term that will allow the expression to be written as the square of a binomial.
We can approach this by trying to form an expression of the type (a + b)^2 = a^2 + 2ab + b^2 where a is given by 2×square (or 2x) and we need to solve for b such that 2ab gives us the middle term, which is ⅔. Knowing that a = 2 in this scenario, we need 2 × 2 × b = ⅔, which simplifies to b = ⅔. Therefore, the missing term or the term b must be ⅙, which is 1 when squared ((⅙)^2 = 1).
The correct answer is therefore, option (c) 1 because adding 1 will complete the perfect square, resulting in the expression being written as (2×square + 1)^2.