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

7 answers

To multiply the polynomials \( (4x^2 + 7x - 3) \) and \( (2x^2 + 7x - 5) \), we can use the distributive property, also known as the FOIL method for binomials, but here we will distribute each term of the first polynomial across each term of the second polynomial.

Let's carry out the multiplication step by step.

  1. Multiply \( 4x^2 \) by each term in the second polynomial: \[ 4x^2 \cdot 2x^2 = 8x^4 \] \[ 4x^2 \cdot 7x = 28x^3 \] \[ 4x^2 \cdot -5 = -20x^2 \]

  2. Multiply \( 7x \) by each term in the second polynomial: \[ 7x \cdot 2x^2 = 14x^3 \] \[ 7x \cdot 7x = 49x^2 \] \[ 7x \cdot -5 = -35x \]

  3. Multiply \( -3 \) by each term in the second polynomial: \[ -3 \cdot 2x^2 = -6x^2 \] \[ -3 \cdot 7x = -21x \] \[ -3 \cdot -5 = 15 \]

  4. Now, combine all these results together: \[ 8x^4 + 28x^3 - 20x^2 + 14x^3 + 49x^2 - 35x - 6x^2 - 21x + 15 \]

  5. Combine like terms:

    • For \( x^4 \): \( 8x^4 \)
    • For \( x^3 \): \( 28x^3 + 14x^3 = 42x^3 \)
    • For \( x^2 \): \( -20x^2 + 49x^2 - 6x^2 = 23x^2 \)
    • For \( x \): \( -35x - 21x = -56x \)
    • Constant: \( 15 \)

Thus, the final result of the multiplication is: \[ \boxed{8x^4 + 42x^3 + 23x^2 - 56x + 15} \]

6x^2+14x-8
2x^2-8
2x^2-14x_8
2x^2+2

It looks like you have provided a list of polynomial expressions, but it's unclear what operation you would like to perform with them. I will interpret it as an instruction to simplify the expressions or do some calculations with them.

Assuming you want to combine like terms or arrange them in a particular way, let's take a look at each expression one by one.

  1. Expression 1: \( 6x^2 + 14x - 8 \)

  2. Expression 2: \( 2x^2 - 8 \)

  3. Expression 3: \( 2x^2 - 14x - 8 \)

  4. Expression 4: \( 2x^2 + 2 \)

If you would like to add or subtract these, we can combine terms directly.

Step 1: Combine Expressions

Let’s combine all four expressions:

\[ (6x^2 + 14x - 8) + (2x^2 - 8) + (2x^2 - 14x - 8) + (2x^2 + 2) \]

Step 2: Group Like Terms

  1. Combine \(x^2\) terms:

    • \(6x^2 + 2x^2 + 2x^2 + 2x^2 = 12x^2\)

  2. Combine \(x\) terms:

    • \(14x - 14x = 0\) (these cancel each other out)

  3. Combine constant terms:

    • \(-8 - 8 - 8 + 2 = -22\)

Final Result

Now, putting everything together:

\[ 12x^2 - 22 \]

Thus, if we add together all the expressions you provided, the combined polynomial is:

\[ \boxed{12x^2 - 22} \]

If you meant something else with your expressions or if there is a specific action you want me to perform (like factoring or evaluating), please clarify!

(4x^2+7x-3)(2x^2+7x-5)
Answer choices
6x^2+14x-8
2x^2-8
2x^2-14x_8
2x^2+2

It appears you want to determine which of the provided answer choices corresponds to the result of multiplying the two polynomials \( (4x^2 + 7x - 3) \) and \( (2x^2 + 7x - 5) \).

We previously calculated the product of the two polynomials, and the result was: \[ 8x^4 + 42x^3 + 23x^2 - 56x + 15 \]

Let's analyze the provided options:

  1. \(6x^2 + 14x - 8\): This is a quadratic polynomial, and its degree does not match the degree of the product we computed.

  2. \(2x^2 - 8\): This is also a quadratic polynomial and cannot match the degree of the product.

  3. \(2x^2 - 14x - 8\): This is again a quadratic polynomial and doesn't match our higher-degree result.

  4. \(2x^2 + 2\): This is also a quadratic polynomial.

None of the answer choices you provided match the degree and the resultant polynomial of \(8x^4 + 42x^3 + 23x^2 - 56x + 15\).

If these are the only options available, it seems that there might be a misunderstanding, or there could be an error in the answer choices. None of them are equivalent to the correct result of multiplying the two polynomials.

If you need further assistance or another task regarding the polynomials, feel free to ask!

given the choices, clearly the user meant (4x^2+7x-3) + (2x^2+7x-5)

Thank you for the clarification! Let's find the sum of the polynomials \( (4x^2 + 7x - 3) \) and \( (2x^2 + 7x - 5) \).

Step 1: Add the polynomials

\[ (4x^2 + 7x - 3) + (2x^2 + 7x - 5) \]

Step 2: Combine like terms

  • For \(x^2\) terms: \[ 4x^2 + 2x^2 = 6x^2 \]

  • For \(x\) terms: \[ 7x + 7x = 14x \]

  • For constant terms: \[ -3 - 5 = -8 \]

Final Result

Putting it all together, we get:

\[ 6x^2 + 14x - 8 \]

Conclusion

The result of adding the two polynomials is: \[ \boxed{6x^2 + 14x - 8} \]

This matches the first choice you provided earlier. Thank you for your patience! If you have any further questions, feel free to ask.