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If m2+p2=xm^{2}+p^{2}=x and 4mp=y4mp=y, which of the following is equivalent to 4x+2y4x+2y?\newlineChoose 11 answer:\newline(A) (2m+p)2(2m+p)^{2}\newline(B) (2m+2p)2(2m+2p)^{2}\newline(C) (4m+2p)2(4m+2p)^{2}\newline(D) (4m+8p)2(4m+8p)^{2}

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Q. If m2+p2=xm^{2}+p^{2}=x and 4mp=y4mp=y, which of the following is equivalent to 4x+2y4x+2y?\newlineChoose 11 answer:\newline(A) (2m+p)2(2m+p)^{2}\newline(B) (2m+2p)2(2m+2p)^{2}\newline(C) (4m+2p)2(4m+2p)^{2}\newline(D) (4m+8p)2(4m+8p)^{2}
  1. Given Equations: We are given two equations:\newline11. m2+p2=xm^2 + p^2 = x\newline22. 4mp=y4mp = y\newlineWe need to find an equivalent expression for 4x+2y4x + 2y using these equations.
  2. Express 4x4x: First, let's express 4x4x in terms of mm and pp using the first equation:\newline4x=4(m2+p2)4x = 4(m^2 + p^2)
  3. Express 2y2y: Now, let's express 2y2y in terms of mm and pp using the second equation:\newline2y=2(4mp)2y = 2(4mp)
  4. Combine Expressions: Combine the expressions for 4x4x and 2y2y: \newline4x+2y=4(m2+p2)+2(4mp)4x + 2y = 4(m^2 + p^2) + 2(4mp)
  5. Simplify Expression: Simplify the expression by distributing the constants: 4x+2y=4m2+4p2+8mp4x + 2y = 4m^2 + 4p^2 + 8mp
  6. Factor Expression: Now, let's try to factor this expression to match one of the answer choices. We are looking for a perfect square since all the answer choices are in the form of a squared binomial.
  7. Identify Perfect Square: We notice that 4m2+4p2+8mp4m^2 + 4p^2 + 8mp can be written as (2m+2p)2(2m + 2p)^2 because:\newline(\(2\)m + \(2\)p)^\(2\) = (\(2\)m)^\(2\) + \(2\)\times(\(2\)m)\times(\(2\)p) + (\(2\)p)^\(2\)\(\newline\) = \(4\)m^\(2\) + \(8\)mp + \(4\)p^\(2\)\newlineWhich matches our expression for 4x+2y4x + 2y.
  8. Final Equivalent Expression: Therefore, the equivalent expression for 4x+2y4x + 2y is (2m+2p)2(2m + 2p)^2, which corresponds to answer choice (B)(B).

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