Question

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Explanation:

Step1: Factor the denominator

First, factor the denominator of the first fraction. The denominator \(4a - 4\) can be factored as \(4(a - 1)\). So the expression becomes \(\frac{4a}{4(a - 1)} \cdot \frac{a - 1}{a + 1}\).

Step2: Cancel out common terms

Now, we can cancel out the common factors. The \(4\) in the numerator and denominator of the first fraction cancels out, and the \((a - 1)\) terms in the numerator of the second fraction and the denominator of the first fraction also cancel out. After canceling, we are left with \(\frac{a}{a + 1}\)? Wait, no, wait. Wait, the first numerator is \(4a\), denominator is \(4(a - 1)\), second numerator is \(a - 1\), denominator is \(a + 1\). So cancel \(4\) ( \(4a/4 = a\) ), cancel \(a - 1\) ( \((a - 1)/(a - 1)=1\) ). So we have \(\frac{a}{1} \cdot \frac{1}{a + 1}=\frac{a}{a + 1}\)? Wait, no, wait, original first fraction: \(\frac{4a}{4(a - 1)}=\frac{a}{a - 1}\) (after canceling 4). Then multiply by \(\frac{a - 1}{a + 1}\). So \(\frac{a}{a - 1} \cdot \frac{a - 1}{a + 1}\). Now cancel \(a - 1\), so we get \(\frac{a}{a + 1}\)? Wait, but let's check the options. Wait, maybe I made a mistake. Wait, the first fraction is \(\frac{4a}{4a - 4}\), which is \(\frac{4a}{4(a - 1)}=\frac{a}{a - 1}\). Then multiply by \(\frac{a - 1}{a + 1}\). So \(\frac{a}{a - 1} \times \frac{a - 1}{a + 1}\). The \(a - 1\) cancels, so we have \(\frac{a}{a + 1}\)? Wait, but looking at the options, one of the options is \(\frac{a}{a + 1}\) (the second option? Wait, the options are: first option \(\frac{4a}{a + 1}\), second \(\frac{a}{a + 1}\), third \(\frac{4}{a + 1}\), fourth \(\frac{1}{a + 1}\). Wait, let's re - do the calculation. Wait, original expression: \(\frac{4a}{4a - 4} \cdot \frac{a - 1}{a + 1}\). Factor \(4a - 4 = 4(a - 1)\), so \(\frac{4a}{4(a - 1)} \cdot \frac{a - 1}{a + 1}\). Now, cancel 4: \(\frac{a}{a - 1} \cdot \frac{a - 1}{a + 1}\). Now cancel \(a - 1\): \(\frac{a}{a + 1}\). So the correct answer should be the option with \(\frac{a}{a + 1}\) (the second option in the vertical list? Wait, the options are shown as: first \(\frac{4a}{a + 1}\), second \(\frac{a}{a + 1}\), third \(\frac{4}{a + 1}\), fourth \(\frac{1}{a + 1}\)).

Answer:

The correct option is the second one (assuming the second box has \(\frac{a}{a + 1}\))