Question

1. \\(\frac{10x^2 - 10}{5x^2 + 2x - 3}\\)

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(10x^2 - 10\). We can factor out a 10: \(10(x^2 - 1)\). Then, using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\), we get \(10(x + 1)(x - 1)\).

Next, factor the denominator \(5x^2+2x - 3\). We need to find two numbers that multiply to \(5\times(-3)=-15\) and add to 2. Those numbers are 5 and -3. So we rewrite the middle term: \(5x^2+5x - 3x - 3\). Then factor by grouping: \(5x(x + 1)-3(x + 1)=(5x - 3)(x + 1)\).

So now the expression is \(\frac{10(x + 1)(x - 1)}{(5x - 3)(x + 1)}\).

Step2: Cancel common factors

We can cancel out the common factor of \((x + 1)\) (assuming \(x
eq - 1\) to avoid division by zero). After canceling, we are left with \(\frac{10(x - 1)}{5x - 3}\). We can also distribute the 10 in the numerator: \(\frac{10x-10}{5x - 3}\), but the simplified form with the factored or distributed numerator is acceptable.

Answer:

\(\frac{10(x - 1)}{5x - 3}\) (or \(\frac{10x - 10}{5x - 3}\))