Question

rational expressions
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divide and state the quotient in simplest form.
\\(\frac{5y}{y^2 + 4y + 4} \div \frac{20y^2}{y^2 - 4}\\)
\\(\circ \frac{1}{4y}\\)
\\(\circ \frac{y^2 - 4}{4y(y^2 + 4y + 4)}\\)
\\(\circ \frac{4y(y + 2)}{y - 2}\\)
\\(\circ \frac{y - 2}{4y^2 + 8y}\\)
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Explanation:

Step1: Recall division of rational expressions

To divide two rational expressions, we multiply the first by the reciprocal of the second. So, \(\frac{5y}{y^{2}+4y + 4}\div\frac{20y^{2}}{y^{2}-4}=\frac{5y}{y^{2}+4y + 4}\times\frac{y^{2}-4}{20y^{2}}\)

Step2: Factor the quadratic expressions

Factor \(y^{2}+4y + 4\) using the perfect square formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), here \(a = y\), \(b = 2\), so \(y^{2}+4y + 4=(y + 2)^{2}\)
Factor \(y^{2}-4\) using the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), here \(a = y\), \(b = 2\), so \(y^{2}-4=(y + 2)(y - 2)\)

Step3: Substitute the factored forms into the expression

Substituting, we get \(\frac{5y}{(y + 2)^{2}}\times\frac{(y + 2)(y - 2)}{20y^{2}}\)

Step4: Simplify the expression

Cancel out common factors. Cancel one \(y\) from numerator and denominator, one \((y + 2)\) from numerator and denominator, and simplify the constants \(\frac{5}{20}=\frac{1}{4}\)
So we have \(\frac{1\times(y - 2)}{4y\times(y + 2)}=\frac{y - 2}{4y(y + 2)}\)
Simplify the denominator \(4y(y + 2)=4y^{2}+8y\)

Answer:

\(\boldsymbol{\frac{y - 2}{4y^{2}+8y}}\) (which corresponds to the last option)