To find the product of the given polynomials [tex]\((4x^2 + 2)(6x^2 + 8x + 5)\)[/tex], we need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms. Here's a step-by-step breakdown:
1. Multiply each term:
- [tex]\(4x^2 \cdot 6x^2 = 24x^4\)[/tex]
- [tex]\(4x^2 \cdot 8x = 32x^3\)[/tex]
- [tex]\(4x^2 \cdot 5 = 20x^2\)[/tex]
- [tex]\(2 \cdot 6x^2 = 12x^2\)[/tex]
- [tex]\(2 \cdot 8x = 16x\)[/tex]
- [tex]\(2 \cdot 5 = 10\)[/tex]
2. Combine like terms:
- For [tex]\(x^4\)[/tex] terms: [tex]\(24x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(32x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(20x^2 + 12x^2 = 32x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(16x\)[/tex]
- For the constant term: [tex]\(10\)[/tex]
So, the product of the polynomials [tex]\( (4x^2 + 2)(6x^2 + 8x + 5) \)[/tex] is:
[tex]\[ 24x^4 + 32x^3 + 32x^2 + 16x + 10 \][/tex]
Thus, the completed equation is:
[tex]\[ 24x^4 + 32x^3 + 32x^2 + 16x + 10 \][/tex]
So, the answer should be:
[tex]\[ 24 \ x^4 + \ 32 \ x^3 + \ 32 \ x^2 + \ 16 \ x + \ 10 \][/tex]
