Let's break down the two problems presented in the image:

Problem 1: Polynomial Division and Remainders

You are given a cubic expression $x^3 + ax^2 + bx + c$ and the following information:

• When divided by $x - 1$, the remainder is 14.
• When divided by $x + 2$, the remainder is 22.
• $x + 1$ is a factor of the expression.

Steps:

1. Use the remainder theorem to substitute values of $x = 1$ and $x = -2$ into the polynomial to get equations for the remainders.

   • For $x - 1$, the remainder is 14: $1^3 + a(1)^2 + b(1) + c = 14 \quad \Rightarrow \quad 1 + a + b + c = 14$ $a + b + c = 13 \quad \text{(Equation 1)}$

   • For $x + 2$, the remainder is 22: $(-2)^3 + a(-2)^2 + b(-2) + c = 22 \quad \Rightarrow \quad -8 + 4a - 2b + c = 22$ $4a - 2b + c = 30 \quad \text{(Equation 2)}$

2. Since $x + 1$ is a factor of the polynomial, substitute $x = -1$ (again using the remainder theorem), and set the expression equal to zero: $(-1)^3 + a(-1)^2 + b(-1) + c = 0 \quad \Rightarrow \quad -1 + a - b + c = 0$ $a - b + c = 1 \quad \text{(Equation 3)}$

Now you have a system of three equations:

1. $a + b + c = 13$
2. $4a - 2b + c = 30$
3. $a - b + c = 1$

By solving this system, you will find the values of $a$, $b$, and $c$.

Problem 2: Tangents to a Circle

You are given a circle with the equation $x^2 + y^2 = a^2$, and you need to find the equation of the tangents to the circle under three conditions:

1. Tangent parallel to the line $y = mx + c$: The slope of a line tangent to a circle is given by the derivative of the circle equation. The slope $m$ of the tangent must be equal to the slope of the line given, which is $m$.

2. Tangent perpendicular to the line $y = mx + c$: For perpendicularity, the product of the slopes of two lines must equal $-1$. Hence, the slope of the tangent line must be $-\frac{1}{m}$.

3. Tangent passing through the point $(b, 0)$: The equation of the tangent line passing through a point on the circle can be found by using the point-slope form of the equation, combined with the geometric properties of tangents.

Would you like me to proceed with the detailed solutions for either or both of these problems?

Here are some further questions related to these concepts:

1. How would you solve a similar system of equations for higher-degree polynomials?
2. Can you explore how the remainder theorem is applied to different divisor forms?
3. What happens when the divisor is not a linear binomial but a quadratic?
4. Can you derive the general formula for tangents to an ellipse, as an extension of circles?
5. How would you find the equation of a secant line instead of a tangent?

Tip: When solving a system of equations, it's often helpful to start by simplifying one of the equations and then substitute the values into the others.