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Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex], we can use the distributive property, also known as the FOIL method when applied to binomials, to expand the expression. Here's how you can do it step-by-step:
1. Distribute each term from the first polynomial to each term of the second polynomial:
[tex]\[
(7x^2 + 5x + 7)(4x - 6) = 7x^2(4x) + 7x^2(-6) + 5x(4x) + 5x(-6) + 7(4x) + 7(-6)
\][/tex]
2. Calculate each term:
- [tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- [tex]\(7x^2 \times (-6) = -42x^2\)[/tex]
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times (-6) = -30x\)[/tex]
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times (-6) = -42\)[/tex]
3. Combine like terms:
Collect the [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex] terms together:
- The [tex]\(x^3\)[/tex] term is [tex]\(28x^3\)[/tex].
- The [tex]\(x^2\)[/tex] terms are [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex].
- The [tex]\(x\)[/tex] terms are [tex]\(-30x + 28x = -2x\)[/tex].
- The constant term is [tex]\(-42\)[/tex].
4. Write the final expression:
Putting it all together, we have:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
So, the correct answer is:
[tex]\[ \boxed{C. \, 28x^3 - 22x^2 - 2x - 42} \][/tex]
1. Distribute each term from the first polynomial to each term of the second polynomial:
[tex]\[
(7x^2 + 5x + 7)(4x - 6) = 7x^2(4x) + 7x^2(-6) + 5x(4x) + 5x(-6) + 7(4x) + 7(-6)
\][/tex]
2. Calculate each term:
- [tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- [tex]\(7x^2 \times (-6) = -42x^2\)[/tex]
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times (-6) = -30x\)[/tex]
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times (-6) = -42\)[/tex]
3. Combine like terms:
Collect the [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex] terms together:
- The [tex]\(x^3\)[/tex] term is [tex]\(28x^3\)[/tex].
- The [tex]\(x^2\)[/tex] terms are [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex].
- The [tex]\(x\)[/tex] terms are [tex]\(-30x + 28x = -2x\)[/tex].
- The constant term is [tex]\(-42\)[/tex].
4. Write the final expression:
Putting it all together, we have:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
So, the correct answer is:
[tex]\[ \boxed{C. \, 28x^3 - 22x^2 - 2x - 42} \][/tex]
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