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Answer :
To solve the problem of expanding the expression [tex]\((2x + 5)(7 - 4x)\)[/tex] and finding which option represents the product, we'll use the distributive property (also known as the FOIL method for binomials). Let's go through the steps:
1. Multiply each term in the first binomial by each term in the second binomial.
- Multiply the first term of the first binomial by the first term of the second binomial:
[tex]\[
2x \times 7 = 14x
\][/tex]
- Multiply the first term of the first binomial by the second term of the second binomial:
[tex]\[
2x \times (-4x) = -8x^2
\][/tex]
- Multiply the second term of the first binomial by the first term of the second binomial:
[tex]\[
5 \times 7 = 35
\][/tex]
- Multiply the second term of the first binomial by the second term of the second binomial:
[tex]\[
5 \times (-4x) = -20x
\][/tex]
2. Combine all the products.
The expression becomes:
[tex]\[
14x + (-8x^2) + 35 + (-20x)
\][/tex]
3. Combine like terms and arrange in standard quadratic form.
- Combine the [tex]\(x\)[/tex]-terms: [tex]\(14x - 20x = -6x\)[/tex]
- The expression in standard quadratic form is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
So, the quadratic expression that represents the product is [tex]\(-8x^2 - 6x + 35\)[/tex].
Let's identify which option corresponds to this expression:
- Option A: [tex]\(-8x^2 + 6x - 35\)[/tex]
- Option B: [tex]\(-8x^2 - 6x + 35\)[/tex]
- Option C: [tex]\(-8x^2 + 34x - 35\)[/tex]
- Option D: [tex]\(-8x^2 - 34x + 35\)[/tex]
The correct option is B: [tex]\(-8x^2 - 6x + 35\)[/tex].
1. Multiply each term in the first binomial by each term in the second binomial.
- Multiply the first term of the first binomial by the first term of the second binomial:
[tex]\[
2x \times 7 = 14x
\][/tex]
- Multiply the first term of the first binomial by the second term of the second binomial:
[tex]\[
2x \times (-4x) = -8x^2
\][/tex]
- Multiply the second term of the first binomial by the first term of the second binomial:
[tex]\[
5 \times 7 = 35
\][/tex]
- Multiply the second term of the first binomial by the second term of the second binomial:
[tex]\[
5 \times (-4x) = -20x
\][/tex]
2. Combine all the products.
The expression becomes:
[tex]\[
14x + (-8x^2) + 35 + (-20x)
\][/tex]
3. Combine like terms and arrange in standard quadratic form.
- Combine the [tex]\(x\)[/tex]-terms: [tex]\(14x - 20x = -6x\)[/tex]
- The expression in standard quadratic form is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
So, the quadratic expression that represents the product is [tex]\(-8x^2 - 6x + 35\)[/tex].
Let's identify which option corresponds to this expression:
- Option A: [tex]\(-8x^2 + 6x - 35\)[/tex]
- Option B: [tex]\(-8x^2 - 6x + 35\)[/tex]
- Option C: [tex]\(-8x^2 + 34x - 35\)[/tex]
- Option D: [tex]\(-8x^2 - 34x + 35\)[/tex]
The correct option is B: [tex]\(-8x^2 - 6x + 35\)[/tex].
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