The given quadratic function is:

$$54x^4 + 216x^2 + 210$$

It is factored in the form:

$$k(ax^2 + b)(cx^2 + d)$$

where $a, b, c, d, k$ are integers. We are tasked to find the smallest possible value of $ab$.

Solution:

First, factor out the greatest common factor (GCF) from the given quadratic:

$$54x^4 + 216x^2 + 210 = 6(9x^4 + 36x^2 + 35)$$

Now, focus on factoring the quadratic expression $9x^4 + 36x^2 + 35$:

$$9x^4 + 36x^2 + 35 = (3x^2 + 7)(3x^2 + 5)$$

So, the complete factorization is:

$$54x^4 + 216x^2 + 210 = 6(3x^2 + 7)(3x^2 + 5)$$

From the factorization, we have:

• $k = 6$
• $a = 3, b = 7$

The product $ab$ is:

$$ab = 3 \cdot 5 = 15$$

Conclusion: The smallest possible value of $ab$ is:

Answer: C) 15