(4x^2+7x-3)(2x^2+7x-5)

Question

What is the expression that models the remainder of Alex's workshop space once he rents Matt his portion?
Answer choices
6x^2+14x-8
2x^2-8
2x^2-14x_8
2x^2+2

Answer 1

To find the expression that models the remainder of Alex's workshop space once he rents Matt his portion, we need to multiply the two polynomials given: \( (4x^2 + 7x - 3)(2x^2 + 7x - 5) \).

Let's perform the multiplication step by step:

1. Use the distributive property (FOIL method): \[ (4x^2)(2x^2) + (4x^2)(7x) + (4x^2)(-5) + (7x)(2x^2) + (7x)(7x) + (7x)(-5) + (-3)(2x^2) + (-3)(7x) + (-3)(-5) \]

2. Calculate each term:

   • \( (4x^2)(2x^2) = 8x^4 \)
   • \( (4x^2)(7x) = 28x^3 \)
   • \( (4x^2)(-5) = -20x^2 \)
   • \( (7x)(2x^2) = 14x^3 \)
   • \( (7x)(7x) = 49x^2 \)
   • \( (7x)(-5) = -35x \)
   • \( (-3)(2x^2) = -6x^2 \)
   • \( (-3)(7x) = -21x \)
   • \( (-3)(-5) = 15 \)

3. Combine all the like terms: \[ 8x^4 + (28x^3 + 14x^3) + (-20x^2 + 49x^2 - 6x^2) + (-35x - 21x) + 15 \] Simplifying gives:

   • \( 8x^4 \)
   • \( 42x^3 \)
   • \( 23x^2 \)
   • \( -56x \)
   • \( +15 \)

So the product is: \[ 8x^4 + 42x^3 + 23x^2 - 56x + 15 \]

None of the provided answer choices represent this expression directly, likely indicating that the question may have been about a specific part of the problem or derived from division by another polynomial.

If we are to find a remainder or a certain aspect related to Alex’s workshop space, we need more information or clarification about the conditions of the question. The polynomial multiplication alone does not directly relate to "the remainder."

Please provide additional context on how to interpret the expression for Alex's workshop space and how it relates to the given options.

Answer 2

Alex has a workshop with an area that can be modeled by the expression 4x^2+7x-3 square feet. He agrees to rent out part of his workshop to his friend Matt. Alex agrees to let Matt use a space in his workshop that is 2x^2+7x-5 square feet.
What is the expression that models the remainder of Alex's workshop space once he rents Matt his portion?
Answer choices
6x^2+14x-8
2x^2-8
2x^2-14x_8
2x^2+2

Answer 3

To determine the remaining area of Alex's workshop after he rents out a portion to Matt, we need to subtract Matt's portion from Alex's total workshop area.

Given:

• Alex's workshop area: \(4x^2 + 7x - 3\)
• Matt's portion: \(2x^2 + 7x - 5\)

The expression for the remaining area is: \[ \text{Remaining area} = \text{Alex's area} - \text{Matt's portion} \] This can be written as: \[ \text{Remaining area} = (4x^2 + 7x - 3) - (2x^2 + 7x - 5) \]

Next, we will distribute the negative sign to the terms in Matt's portion and combine like terms: \[ = 4x^2 + 7x - 3 - 2x^2 - 7x + 5 \]

Now, combine the like terms:

1. Combine the \(x^2\) terms: \[ 4x^2 - 2x^2 = 2x^2 \]

2. Combine the \(x\) terms: \[ 7x - 7x = 0 \]

3. Combine the constant terms: \[ -3 + 5 = 2 \]

Putting it all together, we have: \[ \text{Remaining area} = 2x^2 + 0 + 2 \] which simplifies to: \[ 2x^2 + 2 \]

Thus, the expression that models the remainder of Alex's workshop space is \(2x^2 + 2\).

From the provided answer choices, the correct one is: 2x^2 + 2.