You need to convert the given standard form of quadratic equation into vertex form, such that:

`y = -3x^2 + 2x + 1`

`y = -3(x^2 - 2/3*x) + 1` (factor out -3 between the first and second terms)

You need to complete the square `x^2 - 2/3*x` using the following formula `(a- b)^2 = a^2 - 2ab + b^2` , such that:

`a^2 = x^2 => a = x`

`-2ab = -2*(1/3)*x => -2x*b = -2*x*(1/3) => b = -1/3`

`b^2 = 1/9`

Hence, you need to complete the square adding `1/9` such that:

`y = -3(x^2 - 2/3*x + 1/9) + 1`

Adding `1/9,` the equation changes, hence, you need to subtract 1/9 to prevent the changing of original equation, such that:

`y = -3(x^2 - 2/3*x + 1/9 - 1/9) + 1`

Converting the expansion `x^2 - 2/3*x + 1/9` into binomial `(x^2 - 1/3)^2` yields:

`y = -3((x^2 - 1/3)^2 - 1/9) + 1`

`y = -3(x^2 - 1/3)^2 + 3/9 + 1 => y = -3(x^2 - 1/3)^2 + 1/3 + 1`

You need to bring the terms `1/3 + 1` to a common denominator, such that:

`1/3 + 1 = (1 + 3)/3 = 4/3 y = -3(x^2 - 1/3)^2 + 4/3`

**Hence, converting the standard form of quadratic equation `y = -3x^2 + 2x + 1` into the vertex form yields `y = -3(x^2 - 1/3)^2 + 4/3.` **