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Answer :
To simplify the expression [tex]\((x^2 + 5x + 3)(-4x + 1)\)[/tex], we need to use the distributive property, also known as the FOIL method when dealing with binomials.
Here are the steps:
1. Distribute each term from the first polynomial [tex]\((x^2 + 5x + 3)\)[/tex] to every term in the second polynomial [tex]\((-4x + 1)\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] by both terms in [tex]\(-4x + 1\)[/tex]:
- [tex]\(x^2 \times -4x = -4x^3\)[/tex]
- [tex]\(x^2 \times 1 = x^2\)[/tex]
3. Multiply [tex]\(5x\)[/tex] by both terms in [tex]\(-4x + 1\)[/tex]:
- [tex]\(5x \times -4x = -20x^2\)[/tex]
- [tex]\(5x \times 1 = 5x\)[/tex]
4. Multiply [tex]\(3\)[/tex] by both terms in [tex]\(-4x + 1\)[/tex]:
- [tex]\(3 \times -4x = -12x\)[/tex]
- [tex]\(3 \times 1 = 3\)[/tex]
5. Combine all the terms you obtained:
- [tex]\(-4x^3 + x^2 - 20x^2 + 5x - 12x + 3\)[/tex]
6. Combine like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(x^2 - 20x^2 = -19x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(5x - 12x = -7x\)[/tex]
7. Write the final expression:
- The simplified form is [tex]\(-4x^3 - 19x^2 - 7x + 3\)[/tex].
Therefore, the simplified expression is [tex]\(-4x^3 - 19x^2 - 7x + 3\)[/tex].
Here are the steps:
1. Distribute each term from the first polynomial [tex]\((x^2 + 5x + 3)\)[/tex] to every term in the second polynomial [tex]\((-4x + 1)\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] by both terms in [tex]\(-4x + 1\)[/tex]:
- [tex]\(x^2 \times -4x = -4x^3\)[/tex]
- [tex]\(x^2 \times 1 = x^2\)[/tex]
3. Multiply [tex]\(5x\)[/tex] by both terms in [tex]\(-4x + 1\)[/tex]:
- [tex]\(5x \times -4x = -20x^2\)[/tex]
- [tex]\(5x \times 1 = 5x\)[/tex]
4. Multiply [tex]\(3\)[/tex] by both terms in [tex]\(-4x + 1\)[/tex]:
- [tex]\(3 \times -4x = -12x\)[/tex]
- [tex]\(3 \times 1 = 3\)[/tex]
5. Combine all the terms you obtained:
- [tex]\(-4x^3 + x^2 - 20x^2 + 5x - 12x + 3\)[/tex]
6. Combine like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(x^2 - 20x^2 = -19x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(5x - 12x = -7x\)[/tex]
7. Write the final expression:
- The simplified form is [tex]\(-4x^3 - 19x^2 - 7x + 3\)[/tex].
Therefore, the simplified expression is [tex]\(-4x^3 - 19x^2 - 7x + 3\)[/tex].
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