a) Factor \( x^{2}+2 x+1 \) :

You can factor \( x^{2}+2x+1 \) into \( (x+1)(x+1) \) or simply write it as \( (x+1)^{2} \). This is a perfect square trinomial, where both roots are the same, which means the graph of this quadratic touches the x-axis at \( x = -1 \). It’s an excellent example of how some polynomials can be neatly packaged into a square! In practical terms, this factoring helps solve quadratic equations efficiently. For instance, if you set \( x^{2} + 2x + 1 = 0 \), you can quickly deduce that \( (x+1)^{2} = 0 \) leads to the solution \( x = -1 \). This not only speeds up your calculations but also provides deeper insights into the behavior of the function!