Instead of multiplying all four terms in each factor, think about treating $x^2 + 3$ as one piece and $a x$ as another. What identity lets you simplify $(P - Q)(P + Q)$ quickly?

After you rewrite and simplify the product, you should get something that looks like $x^4 + (\text{something})x^2 + 9$. Compare that $x^2$ coefficient to $-10$ to form an equation for $a$.

When you solve for $a^2$, you will get two possible values for $a$. Use the fact that $a$ is given to be positive to choose the correct one.

Type f(x) = x^4 - 10x^2 + 9 to graph the given polynomial.

For each option, type a separate expression:

• g2(x) = (x^2 - 2x + 3)(x^2 + 2x + 3)
• g3(x) = (x^2 - 3x + 3)(x^2 + 3x + 3)
• g4(x) = (x^2 - 4x + 3)(x^2 + 4x + 3)
• g5(x) = (x^2 - 5x + 3)(x^2 + 5x + 3)

Look at which graph $g2$, $g3$, $g4$, or $g5$ lies exactly on top of $f(x)$ for all visible $x$-values. The corresponding value of $a$ is the correct choice.

Notice that

has the form $(P - Q)(P + Q)$ where $P = x^2 + 3$ and $Q = a x$.

Using the identity $(P - Q)(P + Q) = P^2 - Q^2$, we can rewrite the product without fully multiplying it out.

Apply $(P - Q)(P + Q) = P^2 - Q^2$ with $P = x^2 + 3$ and $Q = a x$:

Now expand each square.

First expand $(x^2 + 3)^2$:

Then subtract $(a x)^2 = a^2 x^2$:

So the product becomes $x^4 + (6 - a^2)x^2 + 9$.

We are told this equals $x^4 - 10x^2 + 9$.

So the coefficient of $x^2$ must match:

Solve for $a^2$:

Since $a$ is positive, $a = 4$, which corresponds to choice C.