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Multiply. \[ (x+5 w+2)(4 x-5 w) \] Simplify. your answer. पI

Ask by Weston Herrera. in the United States
Jan 27,2025

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Answer

The simplified expression is \(4x^2 + 15wx - 25w^2 + 8x - 10w\).

Solution

Sure, let's multiply and simplify the expression: \[ (x + 5w + 2)(4x - 5w) \] ### Step 1: Distribute Each Term Multiply each term in the first polynomial by each term in the second polynomial: \[ \begin{align*} x \cdot 4x &= 4x^2 \\ x \cdot (-5w) &= -5wx \\ 5w \cdot 4x &= 20wx \\ 5w \cdot (-5w) &= -25w^2 \\ 2 \cdot 4x &= 8x \\ 2 \cdot (-5w) &= -10w \\ \end{align*} \] ### Step 2: Combine Like Terms Now, add all the products together and combine like terms: \[ 4x^2 - 5wx + 20wx - 25w^2 + 8x - 10w \] Combine the \( wx \) terms: \[ -5wx + 20wx = 15wx \] So, the expression simplifies to: \[ 4x^2 + 15wx - 25w^2 + 8x - 10w \] ### Final Simplified Form \[ 4x^2 + 15wx - 25w^2 + 8x - 10w \]

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Beyond the Answer

To multiply the expressions \((x+5)(w+2)(4x-5w)\), we'll apply the distributive property, also known as the FOIL method for binomials. First, distribute \((x + 5)(w + 2)\): \[ = x(w + 2) + 5(w + 2) \\ = xw + 2x + 5w + 10 \] Now, we have \(xw + 2x + 5w + 10\). Next, we'll multiply this result by \((4x - 5w)\): \[ = (xw + 2x + 5w + 10)(4x - 5w) \] Distributing \(4x\) and \(-5w\) to each term: \[ = 4x(xw) + 4x(2x) + 4x(5w) + 4x(10) - 5w(xw) - 5w(2x) - 5w(5w) - 5w(10) \] This results in: \[ = 4x^2w + 8x^2 + 20x + 40x - 5w^2x - 10wx - 25w^2 - 50w \] Now, combine like terms: \[ = (4x^2w - 5w^2x) + (8x^2 - 10wx) + (20x - 25w^2 - 50w) \] Putting together common terms, our final expression simplifies to: \[ = 4x^2w - 5w^2x + 8x^2 - 10wx + 20x - 25w^2 - 50w \] This is a simplified version of the product \((x+5)(w+2)(4x-5w)\).

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