Decoding the Quadratic Puzzle: Factor f(x) = x² - 6x + 9

To solve this problem, we need to express the quadratic function $f(x) = x^2 - 6x + 9$ as a product of binomials.

First, observe whether the expression can be written as a perfect square trinomial. It helps to compare it with the standard perfect square form: $(x-a)^2 = x^2 - 2ax + a^2$.

In our quadratic, we have:

• The quadratic term $x^2$ matches exactly.
• The linear term $-6x$ suggests $-2a = -6$. Solving for $a$, we have $a = 3$. Hence, the expression of vertex form should be $(x-3)^2$.
• The constant term is $9$. In a perfect square trinomial, this term would also be $3^2 = 9$, which fits perfectly.

Thus, the original quadratic function $x^2 - 6x + 9$ can be rewritten as a squared binomial: $(x-3)^2$.

Our detailed work confirms that the representation of the function is $(x-3)^2$. This matches with choice 4.

Therefore, the product representation of the function is $\boldsymbol{(x-3)^2}$.