c) \( \left(2 x^{3} y^{2}-3 x^{2} y^{4}+x y\right)(5 x-2) \)
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The Deep Dive
To multiply the expression \( \left(2 x^{3} y^{2}-3 x^{2} y^{4}+x y\right)(5 x-2) \), we can use the distributive property, also known as the FOIL method (First, Outside, Inside, Last) for polynomials. Here's how it goes: First, distribute \(5x\) over each term in the polynomial: 1. \(5x \cdot 2x^3y^2 = 10x^4y^2\) 2. \(5x \cdot (-3x^2y^4) = -15x^3y^4\) 3. \(5x \cdot xy = 5x^2y\) Next, distribute \(-2\) over each term in the polynomial: 1. \(-2 \cdot 2x^3y^2 = -4x^3y^2\) 2. \(-2 \cdot (-3x^2y^4) = 6x^2y^4\) 3. \(-2 \cdot xy = -2xy\) Now, combine all the terms together: \[ 10x^4y^2 + (-15x^3y^4) + 5x^2y + (-4x^3y^2) + 6x^2y^4 - 2xy \] Now, we group like terms: - The \(x^4y^2\) term: \(10x^4y^2\) - The \(x^3y^4\) terms: \(-15x^3y^4 - 4x^3y^2 = -19x^3y^2\) - The \(x^2y\) term: \(5x^2y + 6x^2y^4 = 11x^2y^4\) - The \(xy\) term: \(-2xy\) Thus, the final expression simplifies to: \[ 10x^4y^2 - 19x^3y^2 + 11x^2y^4 - 2xy \]