a perfect square trinomial can be represented by a square model with equivalent length and width. which…

Explanation:

Step1: Recall perfect - square trinomial formula

The formula for a perfect - square trinomial is $(a\pm b)^2=a^{2}\pm2ab + b^{2}$.

Step2: Check option $x^{2}-6x + 9$

For the polynomial $x^{2}-6x + 9$, we have $a = x$, and $2ab=6x$. If $a = x$, then $2b = 6$, so $b = 3$. And $x^{2}-6x + 9=x^{2}-2\times x\times3+3^{2}=(x - 3)^{2}$.

Step3: Check option $x^{2}-2x + 4$

For the polynomial $x^{2}-2x + 4$, if $a=x$, then $2ab = 2x$, so $b = 1$, but $b^{2}=1\neq4$. So it is not a perfect - square trinomial.

Step4: Check option $x^{2}+5x + 10$

For the polynomial $x^{2}+5x + 10$, if $a = x$, then $2ab=5x$, so $b=\frac{5}{2}$, and $b^{2}=\frac{25}{4}\neq10$. So it is not a perfect - square trinomial.

Step5: Check option $x^{2}+4x + 16$

For the polynomial $x^{2}+4x + 16$, if $a = x$, then $2ab = 4x$, so $b = 2$, but $b^{2}=4\neq16$. So it is not a perfect - square trinomial.

Answer:

$x^{2}-6x + 9$