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What is the simplest form of this expression?

[tex]\[ (x-3)\left(x^2+4x+5\right) \][/tex]

A. [tex]\[ x^3+7x^2+7x+15 \][/tex]

B. [tex]\[ x^3+x^2+17x+15 \][/tex]

C. [tex]\[ x^3+x^2-7x-15 \][/tex]

D. [tex]\[ x^3-7x^2-17x-15 \][/tex]

Sagot :

GinnyAnsweridn
To simplify the expression [tex]\((x-3)(x^2 + 4x + 5)\)[/tex], let's go through the steps methodically.

1. Distribute [tex]\(x-3\)[/tex] across [tex]\(x^2 + 4x + 5\)[/tex]:

[tex]\[ (x-3)(x^2 + 4x + 5) \][/tex]

2. Expand using the distributive property (also known as the FOIL method for binomials):

[tex]\[ = x \cdot (x^2 + 4x + 5) - 3 \cdot (x^2 + 4x + 5) \][/tex]

3. Distribute [tex]\(x\)[/tex] and [tex]\(-3\)[/tex] across the terms inside the parentheses:

[tex]\[ = x \cdot x^2 + x \cdot 4x + x \cdot 5 - 3 \cdot x^2 - 3 \cdot 4x - 3 \cdot 5 \][/tex]

4. Simplify each term:

[tex]\[ = x^3 + 4x^2 + 5x - 3x^2 - 12x - 15 \][/tex]

5. Combine like terms:

[tex]\[ = x^3 + (4x^2 - 3x^2) + (5x - 12x) - 15 \][/tex]

[tex]\[ = x^3 + x^2 - 7x - 15 \][/tex]

So the simplified form of the expression [tex]\((x-3)(x^2 + 4x + 5)\)[/tex] is:

[tex]\[ x^3 + x^2 - 7x - 15 \][/tex]

Checking the answer options:

- A. [tex]\(x^3 + 7x^2 + 7x + 15\)[/tex]
- B. [tex]\(x^3 + x^2 + 17x + 15\)[/tex]
- C. [tex]\(x^3 + x^2 - 7x - 15\)[/tex]
- D. [tex]\(x^3 - 7x^2 - 17x - 15\)[/tex]

Clearly, the simplified expression matches option C.

Therefore, the correct answer is:

C. [tex]\(x^3 + x^2 - 7x - 15\)[/tex]
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