Answer :
To find the product of [tex]\((4x^2 + 6)(x^2 - 3x + 8)\)[/tex], we will use the distributive property, also known as the FOIL method for polynomials, which involves multiplying each term in the first polynomial by each term in the second polynomial. Here’s a step-by-step guide:
1. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(x^2 - 3x + 8\)[/tex]:
- [tex]\(4x^2 \cdot x^2 = 4x^4\)[/tex]
- [tex]\(4x^2 \cdot (-3x) = -12x^3\)[/tex]
- [tex]\(4x^2 \cdot 8 = 32x^2\)[/tex]
2. Multiply [tex]\(6\)[/tex] by each term in [tex]\(x^2 - 3x + 8\)[/tex]:
- [tex]\(6 \cdot x^2 = 6x^2\)[/tex]
- [tex]\(6 \cdot (-3x) = -18x\)[/tex]
- [tex]\(6 \cdot 8 = 48\)[/tex]
3. Add all the terms together:
Combine all the terms you obtained from when you multiplied the polynomials:
[tex]\[
4x^4 - 12x^3 + 32x^2 + 6x^2 - 18x + 48
\][/tex]
4. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(32x^2 + 6x^2 = 38x^2\)[/tex]
Now, our expression looks like:
[tex]\[
4x^4 - 12x^3 + 38x^2 - 18x + 48
\][/tex]
The final expanded product is:
[tex]\[
4x^4 - 12x^3 + 38x^2 - 18x + 48
\][/tex]
Thus, the answer is option B: [tex]\(4x^4 - 12x^3 + 38x^2 - 18x + 48\)[/tex].
1. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(x^2 - 3x + 8\)[/tex]:
- [tex]\(4x^2 \cdot x^2 = 4x^4\)[/tex]
- [tex]\(4x^2 \cdot (-3x) = -12x^3\)[/tex]
- [tex]\(4x^2 \cdot 8 = 32x^2\)[/tex]
2. Multiply [tex]\(6\)[/tex] by each term in [tex]\(x^2 - 3x + 8\)[/tex]:
- [tex]\(6 \cdot x^2 = 6x^2\)[/tex]
- [tex]\(6 \cdot (-3x) = -18x\)[/tex]
- [tex]\(6 \cdot 8 = 48\)[/tex]
3. Add all the terms together:
Combine all the terms you obtained from when you multiplied the polynomials:
[tex]\[
4x^4 - 12x^3 + 32x^2 + 6x^2 - 18x + 48
\][/tex]
4. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(32x^2 + 6x^2 = 38x^2\)[/tex]
Now, our expression looks like:
[tex]\[
4x^4 - 12x^3 + 38x^2 - 18x + 48
\][/tex]
The final expanded product is:
[tex]\[
4x^4 - 12x^3 + 38x^2 - 18x + 48
\][/tex]
Thus, the answer is option B: [tex]\(4x^4 - 12x^3 + 38x^2 - 18x + 48\)[/tex].
