I want all the above solutions in English

Here are the solutions for the questions from the image, in English.

1. (i) Factorization of $f(x)$

Step 1: Given that $(x+1)$ is a factor of $f(x) = x^3 + 6x^2 + 11x + 6$, we perform polynomial division or synthetic division. Using synthetic division with the root $x = -1$:

$$\begin{array}{c|cccc} -1 & 1 & 6 & 11 & 6 \\ & & -1 & -5 & -6 \\ \hline & 1 & 5 & 6 & 0 \\ \end{array}$$

The quotient is $x^2 + 5x + 6$.

Step 2: Factor the quadratic quotient $x^2 + 5x + 6$.
We look for two numbers that sum to 5 and multiply to 6. These numbers are 2 and 3.

$$x^2 + 5x + 6 = (x+2)(x+3)$$

Step 3: Write the complete factorization of $f(x)$.

$$f(x) = (x+1)(x^2 + 5x + 6) = (x+1)(x+2)(x+3)$$

The complete factorization of $f(x)$ is $\boxed{(x+1)(x+2)(x+3)}$.

1. (ii) Show that the roots of the quadratic equation are always real

For a quadratic equation $ax^2 + bx + c = 0$ to have real roots, its discriminant $\Delta = b^2 - 4ac$ must be greater than or equal to zero ($\Delta \ge 0$).

Step 1: Identify the coefficients $a, b, c$ from the equation $3x^2 + (-4 - 2\lambda)x + 2\lambda = 0$.

$$a = 3$$ $$b = (-4 - 2\lambda)$$ $$c = 2\lambda$$

Step 2: Calculate the discriminant $\Delta$.

$$\Delta = b^2 - 4ac$$ $$\Delta = (-4 - 2\lambda)^2 - 4(3)(2\lambda)$$ $$\Delta = (-(4 + 2\lambda))^2 - 24\lambda$$ $$\Delta = (4 + 2\lambda)^2 - 24\lambda$$ $$\Delta = (16 + 16\lambda + 4\lambda^2) - 24\lambda$$ $$\Delta = 4\lambda^2 - 8\lambda + 16$$

Step 3: Show that $\Delta \ge 0$.
Factor out 4 from the expression and complete the square for the quadratic term in $\lambda$:

$$\Delta = 4(\lambda^2 - 2\lambda + 4)$$ $$\Delta = 4((\lambda^2 - 2\lambda + 1) + 3)$$ $$\Delta = 4((\lambda - 1)^2 + 3)$$

Since $(\lambda - 1)^2 \ge 0$ for any real value of $\lambda$, it follows that $(\lambda - 1)^2 + 3 \ge 3$.
Therefore, $\Delta = 4((\lambda - 1)^2 + 3) \ge 4(3) = 12$.
Since $\Delta \ge 12$, the discriminant is always positive, which means the roots of the quadratic equation are always real.

2. (i) Differentiation and equations of the tangent and normal