To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials:
\((12x^2 - 3)(4y^3 + 5x^2)\).
We'll use the distributive property (also known as the FOIL method for binomials) for this multiplication:
- Multiply each term in the first polynomial by each term in the second polynomial.
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)
- \( -3 \cdot 4y^3 = -12y^3 \)
- \( -3 \cdot 5x^2 = -15x^2 \)
- Combine all the results:
\[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
- Rearrange terms in standard polynomial order (usually descending powers of x):
\[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
The final polynomial results from the multiplication of the two given polynomials is:
\[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
None of the original response options exactly match the calculated result. However, all terms are formed by multiplication and demonstrate that the product is also a polynomial, showing the Closure Property of multiplication of polynomials.
If you have to choose from the options provided and notice any simplifications or expressions, please clarify the responses to help me assist you better.