You do not have to expand every term in detail; focus on writing the general form of the product and then grouping like terms to see the coefficients of $x^3$ and $x^2$ in terms of $a$ and $b$.

Once you have the expanded left-hand side, match the coefficient of $x^3$ to get an equation involving $a + b$, and match the coefficient of $x^2$ to get an equation involving $ab$. Use the equation with $ab$ to answer the question.

Even if you can find both $a$ and $b$, the question only asks for $ab$. After setting up the equation involving $ab$, solve it directly instead of spending extra time finding each variable separately.

In Desmos, enter f(x) = (x^2 + a*x + 1)(x^2 + b*x + 4) and g(x) = x^4 + 7x^3 + 17x^2 + 16x + 4. Treat $a$ and $b$ as numbers you will later replace with the values you find algebraically.

After you solve for $a$ and $b$ on paper, substitute those numbers for $a$ and $b$ in the definition of f(x) in Desmos. Check that the graph of f(x) lies exactly on top of the graph of g(x) for all shown $x$; if they coincide everywhere, your value of $ab$ is correct.

Multiply $(x^2 + ax + 1)(x^2 + bx + 4)$ term by term.

• From $x^2 \cdot (x^2 + bx + 4)$ you get $x^4 + bx^3 + 4x^2$.
• From $ax \cdot (x^2 + bx + 4)$ you get $ax^3 + abx^2 + 4ax$.
• From $1 \cdot (x^2 + bx + 4)$ you get $x^2 + bx + 4$.

Now combine like terms.

Add the results from Step 1, grouping by powers of $x$:

This must equal the right-hand side:

Since the polynomials are equal for all $x$, the coefficients of each power of $x$ must match.

• Coefficient of $x^3$: $a + b = 7$.
• Coefficient of $x^2$: $ab + 5 = 17$.

You are asked for $ab$, so focus on the equation involving $ab$.

From the $x^2$-coefficient equation

subtract $5$ from both sides:

So, the value of $ab$ is 12.