Answer :
- Multiply the factors $(2x+5)(7-4x)$ using the distributive property.
- Simplify the expression: $14x - 8x^2 + 35 - 20x$.
- Combine like terms: $-8x^2 - 6x + 35$.
- The resulting quadratic expression is $\boxed{-8 x^2-6 x+35}$.
### Explanation
1. Understanding the Problem
We are given two factors, $(2x+5)$ and $(7-4x)$, and we need to find the quadratic expression that represents their product. This involves multiplying the two factors together.
2. Multiplying the Factors
To find the product of the two factors, we use the distributive property (also known as the FOIL method). This means we multiply each term in the first factor by each term in the second factor:
$(2x+5)(7-4x) = 2x * 7 + 2x * (-4x) + 5 * 7 + 5 * (-4x)$
3. Simplifying the Expression
Now, we simplify each term:
$2x * 7 = 14x$
$2x * (-4x) = -8x^2$
$5 * 7 = 35$
$5 * (-4x) = -20x$
So the expression becomes: $14x - 8x^2 + 35 - 20x$
4. Combining Like Terms
Next, we combine like terms. The like terms are the terms with the same variable and exponent. In this case, we have $14x$ and $-20x$. Combining these gives us:
$14x - 20x = -6x$
So the expression becomes: $-8x^2 - 6x + 35$
5. Comparing with Options
Finally, we compare our resulting quadratic expression, $-8x^2 - 6x + 35$, with the given options. We see that it matches option B.
### Examples
Understanding how to multiply binomials like $(2x+5)$ and $(7-4x)$ is fundamental in many areas, such as calculating the area of a rectangle where the sides are expressed as binomials, or in physics when dealing with projectile motion where the position is described by quadratic expressions. For instance, if you're designing a rectangular garden where the length is $(2x+5)$ meters and the width is $(7-4x)$ meters, multiplying these binomials gives you the area of the garden as a quadratic expression in terms of $x$.
- Simplify the expression: $14x - 8x^2 + 35 - 20x$.
- Combine like terms: $-8x^2 - 6x + 35$.
- The resulting quadratic expression is $\boxed{-8 x^2-6 x+35}$.
### Explanation
1. Understanding the Problem
We are given two factors, $(2x+5)$ and $(7-4x)$, and we need to find the quadratic expression that represents their product. This involves multiplying the two factors together.
2. Multiplying the Factors
To find the product of the two factors, we use the distributive property (also known as the FOIL method). This means we multiply each term in the first factor by each term in the second factor:
$(2x+5)(7-4x) = 2x * 7 + 2x * (-4x) + 5 * 7 + 5 * (-4x)$
3. Simplifying the Expression
Now, we simplify each term:
$2x * 7 = 14x$
$2x * (-4x) = -8x^2$
$5 * 7 = 35$
$5 * (-4x) = -20x$
So the expression becomes: $14x - 8x^2 + 35 - 20x$
4. Combining Like Terms
Next, we combine like terms. The like terms are the terms with the same variable and exponent. In this case, we have $14x$ and $-20x$. Combining these gives us:
$14x - 20x = -6x$
So the expression becomes: $-8x^2 - 6x + 35$
5. Comparing with Options
Finally, we compare our resulting quadratic expression, $-8x^2 - 6x + 35$, with the given options. We see that it matches option B.
### Examples
Understanding how to multiply binomials like $(2x+5)$ and $(7-4x)$ is fundamental in many areas, such as calculating the area of a rectangle where the sides are expressed as binomials, or in physics when dealing with projectile motion where the position is described by quadratic expressions. For instance, if you're designing a rectangular garden where the length is $(2x+5)$ meters and the width is $(7-4x)$ meters, multiplying these binomials gives you the area of the garden as a quadratic expression in terms of $x$.
