the image shows a geometric representation of the function f(x)=x² + 2x + 3 written in standard form. what…

Explanation:

Step1: Recall vertex - form formula

The vertex - form of a quadratic function is $f(x)=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. For the quadratic function $y = ax^{2}+bx + c$, we can complete the square. Given $f(x)=x^{2}+2x + 3$, where $a = 1$, $b = 2$, $c = 3$.

Step2: Complete the square for the $x^{2}+2x$ part

We know that $(x + m)^2=x^{2}+2mx+m^{2}$. For $x^{2}+2x$, if $2m = 2$, then $m = 1$ and $x^{2}+2x=(x + 1)^{2}-1$.

Step3: Rewrite the function

Substitute $x^{2}+2x=(x + 1)^{2}-1$ into $f(x)=x^{2}+2x + 3$. So $f(x)=(x + 1)^{2}-1+3$.

Step4: Simplify the expression

$f(x)=(x + 1)^{2}+2$.

Answer:

$f(x)=(x + 1)^{2}+2$