Once you have the expanded form, line it up with $x^4 - 5x^3 + kx^2 + 45x - 54$. Match the coefficients of $x^3$ and $x$ to create equations involving $a$ and $b$, and match the $x^2$ coefficient to express $k$ in terms of $a$ and $b$.

From the $x^3$ and $x$ coefficients you should get two linear equations in $a$ and $b$. Use substitution or elimination to find $a$ and $b$, then plug them into your expression for the $x^2$ coefficient to get $k$.

In one expression line, type (x^2 - a x + 6)(x^2 - b x - 9). Desmos will automatically expand this and show the coefficients in terms of a and b (including the coefficient of x^2).

In two new lines, type a + b = 5 and 9a - 6b = 45. Desmos will plot these as lines in the (a,b)-plane; look for their intersection point to read the values of a and b.

Go back to the expanded product from step 1 and substitute the intersection values of a and b (or just update the sliders). The coefficient of x^2 in that expanded expression is exactly the value of k; read that number from Desmos.

Write and expand the product:

So the polynomial in factored form is equivalent to

We are told this equals

Because the polynomials are equal for all $x$, the coefficients of each power of $x$ must match:

• $x^3$ term: $-(a+b) = -5$, so $a + b = 5$.
• $x$ term: $9a - 6b = 45$.
• $x^2$ term: $ab - 3 = k$ (this is the expression we will use to find $k$).

The constant term $-54$ already matches automatically.

Use the two equations involving $a$ and $b$:

• $a + b = 5$
• $9a - 6b = 45$

From $a + b = 5$, write $b = 5 - a$ and substitute into the second equation:

Then

Recall the relationship from the $x^2$ term:

Substitute $a = 5$ and $b = 0$:

So the value of $k$ is $-3$.