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Answer :
Let's solve the given problem by expanding the expression [tex]\((2^2 - 4x + 5)(x^2 - 3x + 2)\)[/tex].
1. First Expression:
- The expression inside the first parentheses is [tex]\(2^2 - 4x + 5\)[/tex].
- Simplifying [tex]\(2^2\)[/tex], we have [tex]\(4\)[/tex], so the expression becomes [tex]\(4 - 4x + 5\)[/tex], which simplifies further to [tex]\(9 - 4x\)[/tex].
2. Second Expression:
- The expression inside the second parentheses is [tex]\(x^2 - 3x + 2\)[/tex].
3. Expand the Expression:
- Multiply each term in the first expression with each term in the second expression.
Let us perform the multiplication:
[tex]\[
(9 - 4x)(x^2 - 3x + 2)
\][/tex]
First, distribute the [tex]\(9\)[/tex]:
- [tex]\(9 \times x^2 = 9x^2\)[/tex]
- [tex]\(9 \times (-3x) = -27x\)[/tex]
- [tex]\(9 \times 2 = 18\)[/tex]
Next, distribute the [tex]\(-4x\)[/tex]:
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
Now, add all these terms together:
[tex]\[
9x^2 - 27x + 18 - 4x^3 + 12x^2 - 8x
\][/tex]
Combine like terms:
1. The [tex]\(x^3\)[/tex] term is [tex]\(-4x^3\)[/tex].
2. The [tex]\(x^2\)[/tex] terms: [tex]\(9x^2 + 12x^2 = 21x^2\)[/tex].
3. The [tex]\(x\)[/tex] terms: [tex]\(-27x - 8x = -35x\)[/tex].
4. The constant term is [tex]\(18\)[/tex].
So, the expanded and simplified expression is:
[tex]\[
-4x^3 + 21x^2 - 35x + 18
\][/tex]
The correct option that matches this expanded expression is not among the given choices in the list, so it looks like there was a mistake in the options provided or an issue in matching. Nonetheless, this is the simplified form of the expression.
1. First Expression:
- The expression inside the first parentheses is [tex]\(2^2 - 4x + 5\)[/tex].
- Simplifying [tex]\(2^2\)[/tex], we have [tex]\(4\)[/tex], so the expression becomes [tex]\(4 - 4x + 5\)[/tex], which simplifies further to [tex]\(9 - 4x\)[/tex].
2. Second Expression:
- The expression inside the second parentheses is [tex]\(x^2 - 3x + 2\)[/tex].
3. Expand the Expression:
- Multiply each term in the first expression with each term in the second expression.
Let us perform the multiplication:
[tex]\[
(9 - 4x)(x^2 - 3x + 2)
\][/tex]
First, distribute the [tex]\(9\)[/tex]:
- [tex]\(9 \times x^2 = 9x^2\)[/tex]
- [tex]\(9 \times (-3x) = -27x\)[/tex]
- [tex]\(9 \times 2 = 18\)[/tex]
Next, distribute the [tex]\(-4x\)[/tex]:
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
Now, add all these terms together:
[tex]\[
9x^2 - 27x + 18 - 4x^3 + 12x^2 - 8x
\][/tex]
Combine like terms:
1. The [tex]\(x^3\)[/tex] term is [tex]\(-4x^3\)[/tex].
2. The [tex]\(x^2\)[/tex] terms: [tex]\(9x^2 + 12x^2 = 21x^2\)[/tex].
3. The [tex]\(x\)[/tex] terms: [tex]\(-27x - 8x = -35x\)[/tex].
4. The constant term is [tex]\(18\)[/tex].
So, the expanded and simplified expression is:
[tex]\[
-4x^3 + 21x^2 - 35x + 18
\][/tex]
The correct option that matches this expanded expression is not among the given choices in the list, so it looks like there was a mistake in the options provided or an issue in matching. Nonetheless, this is the simplified form of the expression.
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