Answer :
To multiply the polynomials [tex]\((4x^2 + 4x + 6)(7x + 5)\)[/tex], we'll use the distributive property to expand the expression step by step.
1. Distribute [tex]\(4x^2\)[/tex] to each term in the second polynomial:
- [tex]\(4x^2 \times 7x = 28x^3\)[/tex]
- [tex]\(4x^2 \times 5 = 20x^2\)[/tex]
2. Distribute [tex]\(4x\)[/tex] to each term in the second polynomial:
- [tex]\(4x \times 7x = 28x^2\)[/tex]
- [tex]\(4x \times 5 = 20x\)[/tex]
3. Distribute [tex]\(6\)[/tex] to each term in the second polynomial:
- [tex]\(6 \times 7x = 42x\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]
4. Combine all the terms gathered from the distributions:
- [tex]\(28x^3\)[/tex] (from step 1)
- [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex] (combine the [tex]\(x^2\)[/tex] terms from steps 1 and 2)
- [tex]\(20x + 42x = 62x\)[/tex] (combine the [tex]\(x\)[/tex] terms from steps 2 and 3)
- [tex]\(30\)[/tex] (from step 3)
5. Write the final expanded polynomial:
- [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex]
So, the correct answer is A. [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].
1. Distribute [tex]\(4x^2\)[/tex] to each term in the second polynomial:
- [tex]\(4x^2 \times 7x = 28x^3\)[/tex]
- [tex]\(4x^2 \times 5 = 20x^2\)[/tex]
2. Distribute [tex]\(4x\)[/tex] to each term in the second polynomial:
- [tex]\(4x \times 7x = 28x^2\)[/tex]
- [tex]\(4x \times 5 = 20x\)[/tex]
3. Distribute [tex]\(6\)[/tex] to each term in the second polynomial:
- [tex]\(6 \times 7x = 42x\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]
4. Combine all the terms gathered from the distributions:
- [tex]\(28x^3\)[/tex] (from step 1)
- [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex] (combine the [tex]\(x^2\)[/tex] terms from steps 1 and 2)
- [tex]\(20x + 42x = 62x\)[/tex] (combine the [tex]\(x\)[/tex] terms from steps 2 and 3)
- [tex]\(30\)[/tex] (from step 3)
5. Write the final expanded polynomial:
- [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex]
So, the correct answer is A. [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].
