Answer :
To find the product of [tex]\( (2x^2 + 3x - 1) \)[/tex] and [tex]\( (3x + 5) \)[/tex], we use the distributive property, also known as the FOIL method for binomials, to multiply each term in the first polynomial by each term in the second polynomial.
Here’s a step-by-step breakdown:
1. Distribute each term in the first polynomial to every term in the second polynomial:
- Multiply the first term from [tex]\( (2x^2 + 3x - 1) \)[/tex] by each term in [tex]\( (3x + 5) \)[/tex]:
- [tex]\( 2x^2 \cdot 3x = 6x^3 \)[/tex]
- [tex]\( 2x^2 \cdot 5 = 10x^2 \)[/tex]
- Multiply the second term from [tex]\( (2x^2 + 3x - 1) \)[/tex] by each term in [tex]\( (3x + 5) \)[/tex]:
- [tex]\( 3x \cdot 3x = 9x^2 \)[/tex]
- [tex]\( 3x \cdot 5 = 15x \)[/tex]
- Multiply the third term from [tex]\( (2x^2 + 3x - 1) \)[/tex] by each term in [tex]\( (3x + 5) \)[/tex]:
- [tex]\(-1 \cdot 3x = -3x\)[/tex]
- [tex]\(-1 \cdot 5 = -5\)[/tex]
2. Combine all the resulting terms:
[tex]\[
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5
\][/tex]
3. Combine like terms (terms with the same power of [tex]\( x \)[/tex]):
- The [tex]\( x^3 \)[/tex] term is [tex]\( 6x^3 \)[/tex].
- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\( 10x^2 + 9x^2 = 19x^2 \)[/tex].
- Combine the [tex]\( x \)[/tex] terms: [tex]\( 15x - 3x = 12x \)[/tex].
- The constant term is [tex]\(-5\)[/tex].
Putting it all together, the expanded expression is:
[tex]\[ 6x^3 + 19x^2 + 12x - 5 \][/tex]
Thus, the product of [tex]\( \left(2x^2 + 3x - 1\right) \)[/tex] and [tex]\( (3x + 5) \)[/tex] is [tex]\( 6x^3 + 19x^2 + 12x - 5 \)[/tex]. The correct answer is option A: [tex]\( 6x^3 + 19x^2 + 12x - 5 \)[/tex].
Here’s a step-by-step breakdown:
1. Distribute each term in the first polynomial to every term in the second polynomial:
- Multiply the first term from [tex]\( (2x^2 + 3x - 1) \)[/tex] by each term in [tex]\( (3x + 5) \)[/tex]:
- [tex]\( 2x^2 \cdot 3x = 6x^3 \)[/tex]
- [tex]\( 2x^2 \cdot 5 = 10x^2 \)[/tex]
- Multiply the second term from [tex]\( (2x^2 + 3x - 1) \)[/tex] by each term in [tex]\( (3x + 5) \)[/tex]:
- [tex]\( 3x \cdot 3x = 9x^2 \)[/tex]
- [tex]\( 3x \cdot 5 = 15x \)[/tex]
- Multiply the third term from [tex]\( (2x^2 + 3x - 1) \)[/tex] by each term in [tex]\( (3x + 5) \)[/tex]:
- [tex]\(-1 \cdot 3x = -3x\)[/tex]
- [tex]\(-1 \cdot 5 = -5\)[/tex]
2. Combine all the resulting terms:
[tex]\[
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5
\][/tex]
3. Combine like terms (terms with the same power of [tex]\( x \)[/tex]):
- The [tex]\( x^3 \)[/tex] term is [tex]\( 6x^3 \)[/tex].
- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\( 10x^2 + 9x^2 = 19x^2 \)[/tex].
- Combine the [tex]\( x \)[/tex] terms: [tex]\( 15x - 3x = 12x \)[/tex].
- The constant term is [tex]\(-5\)[/tex].
Putting it all together, the expanded expression is:
[tex]\[ 6x^3 + 19x^2 + 12x - 5 \][/tex]
Thus, the product of [tex]\( \left(2x^2 + 3x - 1\right) \)[/tex] and [tex]\( (3x + 5) \)[/tex] is [tex]\( 6x^3 + 19x^2 + 12x - 5 \)[/tex]. The correct answer is option A: [tex]\( 6x^3 + 19x^2 + 12x - 5 \)[/tex].
