Question

1. \\(\frac{a}{a + 2} - \frac{a + 10}{a^2 - 4}\\)

Explanation:

Step1: Factor the denominator

Notice that \(a^2 - 4\) is a difference of squares, so \(a^2 - 4=(a + 2)(a - 2)\).

Step2: Find a common denominator

The common denominator of \(\frac{a}{a + 2}\) and \(\frac{a + 10}{a^2 - 4}\) is \((a + 2)(a - 2)\). Rewrite \(\frac{a}{a + 2}\) with the common denominator: \(\frac{a(a - 2)}{(a + 2)(a - 2)}\).

Step3: Subtract the fractions

Now we have \(\frac{a(a - 2)}{(a + 2)(a - 2)}-\frac{a + 10}{(a + 2)(a - 2)}=\frac{a(a - 2)-(a + 10)}{(a + 2)(a - 2)}\).

Step4: Expand and simplify the numerator

Expand \(a(a - 2)\) to get \(a^2-2a\). Then the numerator becomes \(a^2-2a-(a + 10)=a^2-2a - a - 10=a^2-3a - 10\).

Step5: Factor the numerator

Factor \(a^2-3a - 10\). We need two numbers that multiply to \(- 10\) and add to \(-3\). Those numbers are \(-5\) and \(2\), so \(a^2-3a - 10=(a - 5)(a + 2)\).

Step6: Simplify the fraction

Now our fraction is \(\frac{(a - 5)(a + 2)}{(a + 2)(a - 2)}\). Cancel out the common factor \((a + 2)\) (assuming \(a
eq - 2\) and \(a
eq2\)) to get \(\frac{a - 5}{a - 2}\).

Answer:

\(\frac{a - 5}{a - 2}\) (for \(a
eq\pm2\))