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Assignment Booklet 3B
ks) 8. Rahim and Kate are factoring $16 x^{2}+4 y^{2}$. Their solutions are as follows:
Rahim
\[
16 x^{2}+4 y^{2}=4\left(4 x^{2}+y^{2}\right)
\]
Kate
\[
\begin{aligned}
16 x^{2}+4 y^{2} & =4\left(4 x^{2}+y^{2}\right) \\
& =4(2 x+y)(2 x-y)
\end{aligned}
\]
Who is correct? Explain your reasoning.
Answer
\(\boxed{\text{Rahim is correct, and Kate is incorrect.}}\)
Step 1 ：First, let's check Rahim's solution by expanding it:
Step 2 ：\(16x^2 + 4y^2 = 4(4x^2 + y^2)\)
Step 3 ：\(16x^2 + 4y^2 = 16x^2 + 4y^2\)
Step 4 ：Rahim's solution matches the original expression.
Step 5 ：Next, let's check Kate's solution by expanding it:
Step 6 ：\(16x^2 + 4y^2 = 4(2x + y)(2x - y)\)
Step 7 ：\(16x^2 + 4y^2 = 16x^2 - 4y^2\)
Step 8 ：Kate's solution does not match the original expression.
Step 9 ：\(\boxed{\text{Rahim is correct, and Kate is incorrect.}}\)
