To find the product of the polynomials [tex]\((3x^2 - 2x - 3)\)[/tex] and [tex]\((5x^2 + 4x + 5)\)[/tex], we need to apply the distributive property (also known as the FOIL method for binomials, extended here for trinomials).
We first write the product in an expanded form:
[tex]\[
(3x^2 - 2x - 3)(5x^2 + 4x + 5)
\][/tex]
Next, we distribute each term from the first polynomial to each term in the second polynomial:
[tex]\[
3x^2(5x^2 + 4x + 5) - 2x(5x^2 + 4x + 5) - 3(5x^2 + 4x + 5)
\][/tex]
Now, we compute the product term-by-term:
[tex]\[
3x^2 \cdot 5x^2 = 15x^4
\][/tex]
[tex]\[
3x^2 \cdot 4x = 12x^3
\][/tex]
[tex]\[
3x^2 \cdot 5 = 15x^2
\][/tex]
[tex]\[
-2x \cdot 5x^2 = -10x^3
\][/tex]
[tex]\[
-2x \cdot 4x = -8x^2
\][/tex]
[tex]\[
-2x \cdot 5 = -10x
\][/tex]
[tex]\[
-3 \cdot 5x^2 = -15x^2
\][/tex]
[tex]\[
-3 \cdot 4x = -12x
\][/tex]
[tex]\[
-3 \cdot 5 = -15
\][/tex]
Now, we combine all these terms together:
[tex]\[
15x^4 + 12x^3 + 15x^2 - 10x^3 - 8x^2 - 10x - 15x^2 - 12x - 15
\][/tex]
Next, we combine like terms:
[tex]\[
15x^4 + (12x^3 - 10x^3) + (15x^2 - 8x^2 - 15x^2) + (-10x - 12x) - 15
\][/tex]
Simplifying further, we get:
[tex]\[
15x^4 + 2x^3 - 8x^2 - 22x - 15
\][/tex]
Thus, the product of the polynomials [tex]\((3x^2 - 2x - 3)\)[/tex] and [tex]\((5x^2 + 4x + 5)\)[/tex] is:
[tex]\[
15x^4 + 2x^3 - 8x^2 - 22x - 15
\][/tex]
So, the correct option is:
A. [tex]\(15x^4 + 2x^3 - 8x^2 - 22x - 15\)[/tex]
