Let $P(x) = 4x^3 - kx^2 - 9x + 6k$. What should $P(3)$ equal if $x - 3$ is a factor, and what equation does that give you for $k$?

After you find $k$, substitute it back into $4x^3 - kx^2 - 9x + 6k$ to get a specific cubic. Then, since you know $x - 3$ is a factor, divide by $x - 3$ to find the quadratic factor and compare with the answer choices.

After you find $k$ algebraically, type the function into Desmos, for example: f(x) = 4x^3 - 27x^2 - 9x + 162 (use your specific value of $k$). This is the original polynomial that should be equal to one of the answer choice expressions for all $x$.

Enter each option as its own function, such as g1(x) = (x - 3)(4x^2 - 15x + 54), g2(x) = (x - 3)(4x^2 + 15x - 54), etc. Compare their graphs to the graph of $f(x)$. The correct choice will have a graph that completely overlaps $f(x)$ for every $x$.

To be precise, for each choice define a difference, like d1(x) = f(x) - g1(x). The correct answer’s difference function will appear as a horizontal line at $y = 0$ for all $x$, showing the two expressions are identical polynomials.

If a polynomial $P(x)$ is divisible by $x - 3$, then $P(3) = 0$.

Let

Substitute $x = 3$:

Compute the numbers:

• $(3)^3 = 27$, so $4(3)^3 = 4 \cdot 27 = 108$.
• $(3)^2 = 9$, so $k(3)^2 = 9k$.
• $9(3) = 27$.

So

Because the polynomial is divisible by $x - 3$, we must have

Solve the equation from Step 1:

Now substitute $k = 27$ back into the polynomial:

Compute the constant term:

so the polynomial becomes

We now need to factor this polynomial, knowing it has a factor of $x - 3$. The other factor will be a quadratic of the form $4x^2 + bx + c$.

Use synthetic division (or long division) to divide $4x^3 - 27x^2 - 9x + 162$ by $x - 3$.

Set up synthetic division with root $3$ and coefficients $4, -27, -9, 162$:

• Bring down $4$.
• Multiply $4$ by $3$ to get $12$; add to $-27$ to get $-15$.
• Multiply $-15$ by $3$ to get $-45$; add to $-9$ to get $-54$.
• Multiply $-54$ by $3$ to get $-162$; add to $162$ to get $0$ (remainder $0$, as expected).

So the quotient is

Therefore,

which matches answer choice D: $(x - 3)(4x^2 - 15x - 54)$.