To solve this problem, we need to use the relationship between the Least Common Multiple (LCM), Highest Common Factor (HCF), and the given polynomials.

We are given:

• LCM of \(p(x)\) and \(q(x)\) is \((x+a)(x^3-a^3)\)
• HCF of \(p(x)\) and \(q(x)\) is \((x^2-ax+a^2)\)
• Polynomial \(p(x) = x^4 + a^2x^2 + a^4\)

And we need to find \(q(x)\).

Using the relationship:

\(LCM(p(x), q(x)) \cdot HCF(p(x), q(x)) = p(x) \cdot q(x).\)

Substitute the given values into the equation:

\(\left[(x+a)(x^3-a^3)\right] \cdot (x^2-ax+a^2) = (x^4 + a^2x^2 + a^4) \cdot q(x).\)

First, let's simplify \((x+a)(x^3-a^3)\):

• Recognize that \(x^3-a^3 = (x-a)(x^2+ax+a^2)\) (using the difference of cubes).
• So, \((x+a)(x^3-a^3) = (x+a)(x-a)(x^2+ax+a^2)\).

Thus, the LCM can be represented as:

\((x+a)(x-a)(x^2+ax+a^2).\)

Now the equation becomes:

\((x+a)(x-a)(x^2+ax+a^2)(x^2-ax+a^2) = (x^4 + a^2x^2 + a^4) \cdot q(x).\)

Simplify the left-hand side:

• \((x+a)(x-a) = x^2-a^2\)
• So, the left side is \((x^2-a^2)(x^2+ax+a^2)(x^2-ax+a^2)\).

With this simplification, we substitute back:

\((x^2-a^2) = q(x),\)

where the root of simplification and cancellation is observed. Thus, the correct polynomial \(q(x)\) that satisfies the equation is:

• \(q(x) = x^2-a^2\)

Therefore, the option that matches is \(x^2-a^2\).