To find the product of the polynomials \( (12x^2 - 3)(4y^3 + 5x^2) \), we will use the distributive property (also known as the FOIL method for binomials).
Here's the step-by-step multiplication:
1. Distribute \( 12x^2 \) to both terms in the second polynomial:
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)
2. Distribute \( -3 \) to both terms in the second polynomial:
- \( -3 \cdot 4y^3 = -12y^3 \)
- \( -3 \cdot 5x^2 = -15x^2 \)
Now, combine all these results:
\[
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
\]
Next, we can write the resulting polynomial in standard form (usually ordered by degree):
\[
60x^4 + 48x^2y^3 - 15x^2 - 12y^3
\]
Now let's check which of the provided responses matches our result:
- The response that matches our result is:
**None of the provided options exactly match the resulting polynomial.**
Therefore, for demonstration purposes, we have calculated the product of the multiplication, which confirms the closure property of multiplication of polynomials; the product is indeed a polynomial.
If there's a requirement for a choice from the options given when none match, I suggest reviewing the options for any potential typographical errors or misinterpretations in the polynomials presented.