12. Draw the graph of \( y=x^{2}+4 x+3 \) for values of \( x \) from -5 to 1. Find: (a) the coordinates of the turning point of the curve, (b) the range of values of \( x \) for which \( y \) in- creases as \( x \) increases, (c) the range of values of \( x \) for which \( y<0 \). (d) the gradient of the curve at the point when (i) \( x=-3.5 \quad \) (ii) -2 (e) On the same axes draw the graph of \( y-x-3=0 \) and then find the roots of the equation \( x^{2}+3 x+1=-2 x-5 \).

To explore the graph \( y = x^2 + 4x + 3 \), we first recognize that it's a quadratic function that forms a parabola. To sketch it for \( x \) values from -5 to 1, we can first calculate some key points. The standard form of a quadratic \( y = ax^2 + bx + c \) helps us identify key characteristics. From it, we find: (a) The turning point, or vertex, of the parabola can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 4 \), so \( x = -\frac{4}{2(1)} = -2 \). Plugging \( x = -2 \) back into the equation gives \( y = 1 \). Hence, the coordinates of the turning point are (-2, 1). (b) The function \( y \) increases when \( x > -2 \) (to the right of the vertex) and decreases when \( x < -2 \). Therefore, the range of \( x \) where \( y \) increases is \( x \in (-2, \infty) \). (c) To find where \( y < 0 \), we need to set up the quadratic equation \( x^2 + 4x + 3 < 0 \). Factoring gives \( (x + 1)(x + 3) < 0 \), which shows that the function is negative between the roots \( x = -3 \) and \( x = -1 \). Thus, the range of \( x \) for which \( y < 0 \) is \( x \in (-3, -1) \). (d) The gradient of the curve is given by the derivative \( y' = 2x + 4 \). Thus: (i) For \( x = -3.5 \), the gradient is \( y'(-3.5) = 2(-3.5) + 4 = -7 + 4 = -3 \). (ii) For \( x = -2 \), \( y'(-2) = 2(-2) + 4 = -4 + 4 = 0 \) (this confirms the turning point). (e) To draw the line \( y - x - 3 = 0 \) or \( y = x + 3 \), simply plot a line with a slope of 1 that crosses the y-axis at 3. To find the roots of the equation \( x^2 + 3x + 1 = -2x - 5 \), we first rearrange it to \( x^2 + 5x + 6 = 0 \) and factor it: \( (x + 2)(x + 3) = 0 \). This gives the roots \( x = -2 \) and \( x = -3 \).