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Answer :
To solve the problem of multiplying the polynomials [tex]\( (8x^2 + 6x + 8) \)[/tex] and [tex]\( (6x - 5) \)[/tex], let's follow these steps:
1. Distribute Each Term in the First Polynomial:
We'll distribute each term in the first polynomial [tex]\( (8x^2 + 6x + 8) \)[/tex] with each term in the second polynomial [tex]\( (6x - 5) \)[/tex]. This involves using the distributive property for multiplication.
2. Multiply Each Pair of Terms:
- Multiply [tex]\( 8x^2 \)[/tex] by each term in [tex]\( (6x - 5) \)[/tex]:
- [tex]\( 8x^2 \times 6x = 48x^3 \)[/tex]
- [tex]\( 8x^2 \times (-5) = -40x^2 \)[/tex]
- Multiply [tex]\( 6x \)[/tex] by each term in [tex]\( (6x - 5) \)[/tex]:
- [tex]\( 6x \times 6x = 36x^2 \)[/tex]
- [tex]\( 6x \times (-5) = -30x \)[/tex]
- Multiply [tex]\( 8 \)[/tex] by each term in [tex]\( (6x - 5) \)[/tex]:
- [tex]\( 8 \times 6x = 48x \)[/tex]
- [tex]\( 8 \times (-5) = -40 \)[/tex]
3. Combine All the Products:
After computing these products, we bring all the terms together:
[tex]\[
48x^3 - 40x^2 + 36x^2 - 30x + 48x - 40
\][/tex]
4. Combine Like Terms:
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\( -40x^2 + 36x^2 = -4x^2 \)[/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\( -30x + 48x = 18x \)[/tex]
5. Write the Final Expression:
After combining like terms, the expression simplifies to:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
This matches Choice A:
[tex]\[ \boxed{48x^3 - 4x^2 + 18x - 40} \][/tex]
1. Distribute Each Term in the First Polynomial:
We'll distribute each term in the first polynomial [tex]\( (8x^2 + 6x + 8) \)[/tex] with each term in the second polynomial [tex]\( (6x - 5) \)[/tex]. This involves using the distributive property for multiplication.
2. Multiply Each Pair of Terms:
- Multiply [tex]\( 8x^2 \)[/tex] by each term in [tex]\( (6x - 5) \)[/tex]:
- [tex]\( 8x^2 \times 6x = 48x^3 \)[/tex]
- [tex]\( 8x^2 \times (-5) = -40x^2 \)[/tex]
- Multiply [tex]\( 6x \)[/tex] by each term in [tex]\( (6x - 5) \)[/tex]:
- [tex]\( 6x \times 6x = 36x^2 \)[/tex]
- [tex]\( 6x \times (-5) = -30x \)[/tex]
- Multiply [tex]\( 8 \)[/tex] by each term in [tex]\( (6x - 5) \)[/tex]:
- [tex]\( 8 \times 6x = 48x \)[/tex]
- [tex]\( 8 \times (-5) = -40 \)[/tex]
3. Combine All the Products:
After computing these products, we bring all the terms together:
[tex]\[
48x^3 - 40x^2 + 36x^2 - 30x + 48x - 40
\][/tex]
4. Combine Like Terms:
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\( -40x^2 + 36x^2 = -4x^2 \)[/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\( -30x + 48x = 18x \)[/tex]
5. Write the Final Expression:
After combining like terms, the expression simplifies to:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
This matches Choice A:
[tex]\[ \boxed{48x^3 - 4x^2 + 18x - 40} \][/tex]
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