Answer :
To find the quadratic expression that represents the product of the factors [tex]\((2x + 5)(7 - 4x)\)[/tex], we can use the distributive property, often referred to as the FOIL method. FOIL stands for First, Outer, Inner, Last, which helps remember which terms to multiply.
Let's go through each step:
1. First: Multiply the first terms in each binomial:
[tex]\[
2x \cdot 7 = 14x
\][/tex]
2. Outer: Multiply the outer terms in the product:
[tex]\[
2x \cdot (-4x) = -8x^2
\][/tex]
3. Inner: Multiply the inner terms:
[tex]\[
5 \cdot 7 = 35
\][/tex]
4. Last: Multiply the last terms in each binomial:
[tex]\[
5 \cdot (-4x) = -20x
\][/tex]
Now that we have multiplied the terms, let's combine them:
- Start with the quadratic term: [tex]\(-8x^2\)[/tex].
- Combine the linear terms: [tex]\(14x\)[/tex] (from the First) and [tex]\(-20x\)[/tex] (from the Last), which simplifies to:
[tex]\[
14x - 20x = -6x
\][/tex]
- Lastly, include the constant term: [tex]\(+35\)[/tex].
Putting it all together, the quadratic expression is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
The expression that matches this calculation is option A: [tex]\(-8x^2 - 6x + 35\)[/tex].
Let's go through each step:
1. First: Multiply the first terms in each binomial:
[tex]\[
2x \cdot 7 = 14x
\][/tex]
2. Outer: Multiply the outer terms in the product:
[tex]\[
2x \cdot (-4x) = -8x^2
\][/tex]
3. Inner: Multiply the inner terms:
[tex]\[
5 \cdot 7 = 35
\][/tex]
4. Last: Multiply the last terms in each binomial:
[tex]\[
5 \cdot (-4x) = -20x
\][/tex]
Now that we have multiplied the terms, let's combine them:
- Start with the quadratic term: [tex]\(-8x^2\)[/tex].
- Combine the linear terms: [tex]\(14x\)[/tex] (from the First) and [tex]\(-20x\)[/tex] (from the Last), which simplifies to:
[tex]\[
14x - 20x = -6x
\][/tex]
- Lastly, include the constant term: [tex]\(+35\)[/tex].
Putting it all together, the quadratic expression is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
The expression that matches this calculation is option A: [tex]\(-8x^2 - 6x + 35\)[/tex].
