1. **State the problem:** We want to find a factor of the polynomial $$60x^4 + 56x^2 + 12$$ of the form $$ax^2 + b$$ where $$a$$ and $$b$$ are positive constants. Then, we need to find a possible value of the product $$ab$$. 2. **Rewrite the polynomial:** Notice that the polynomial is in terms of $$x^4$$ and $$x^2$$. Let $$y = x^2$$, then the polynomial becomes: $$60y^2 + 56y + 12$$ 3. **Factor the quadratic in $$y$$:** We look for factors of the form $$(ay + b)(cy + d)$$ such that: $$ac = 60, \quad bd = 12, \quad ad + bc = 56$$ 4. **Find factor pairs:** - Factors of 60: (1,60), (2,30), (3,20), (4,15), (5,12), (6,10) - Factors of 12: (1,12), (2,6), (3,4) 5. **Try pairs to satisfy middle term 56:** Try $$(a,c) = (6,10)$$ and $$(b,d) = (2,6)$$: $$ad + bc = 6 \times 6 + 10 \times 2 = 36 + 20 = 56$$ which matches. 6. **So the factorization is:** $$60y^2 + 56y + 12 = (6y + 2)(10y + 6)$$ 7. **Rewrite back in terms of $$x$$:** $$(6x^2 + 2)(10x^2 + 6)$$ 8. **Check the factors:** Both factors are of the form $$ax^2 + b$$ with positive $$a$$ and $$b$$. 9. **Calculate possible values of $$ab$$:** - For $$6x^2 + 2$$, $$a=6$$ and $$b=2$$, so $$ab = 12$$ - For $$10x^2 + 6$$, $$a=10$$ and $$b=6$$, so $$ab = 60$$ 10. **Check the options:** The possible values of $$ab$$ from the factors are 12 and 60, but these are not in the options. 11. **Simplify factors by dividing by common factors:** $$6x^2 + 2 = 2(3x^2 + 1)$$ and $$10x^2 + 6 = 2(5x^2 + 3)$$ 12. **Try the factors $$3x^2 + 1$$ and $$5x^2 + 3$$:** - $$ab$$ for $$3x^2 + 1$$ is $$3 \times 1 = 3$$ - $$ab$$ for $$5x^2 + 3$$ is $$5 \times 3 = 15$$ 13. **Check if $$3x^2 + 1$$ and $$5x^2 + 3$$ are factors:** Multiply: $$ (3x^2 + 1)(5x^2 + 3) = 15x^4 + 9x^2 + 5x^2 + 3 = 15x^4 + 14x^2 + 3 $$ 14. **Compare with original polynomial:** Original is $$60x^4 + 56x^2 + 12$$, which is 4 times the above: $$4(15x^4 + 14x^2 + 3) = 60x^4 + 56x^2 + 12$$ 15. **Therefore, the factor $$4(3x^2 + 1)(5x^2 + 3)$$ equals the original polynomial. So the factors $$3x^2 + 1$$ and $$5x^2 + 3$$ are factors of the polynomial up to a constant multiple. Since the problem asks for a factor of the form $$ax^2 + b$$ with positive constants, $$5x^2 + 3$$ is a valid factor. 16. **Hence, a possible value of $$ab$$ is:** $$5 \times 3 = 15$$ **Final answer:** 15