Let's factor each of the given polynomials step by step:

1. $24x^2 - 22x - 35$

We can factor this quadratic by using the method of grouping. First, we need to find two numbers that multiply to $24 \times -35 = -840$ and add to $-22$.

The numbers that work are $-42$ and $20$, so we rewrite the middle term: $24x^2 - 42x + 20x - 35$ Now, group the terms: $(24x^2 - 42x) + (20x - 35)$ Factor each group: $6x(4x - 7) + 5(4x - 7)$ Factor out the common binomial factor: $(4x - 7)(6x + 5)$

2. $2x^3 + 3x^2 - 17x - 30$

We can use synthetic division or grouping. Let’s try grouping: Group terms: $(2x^3 + 3x^2) + (-17x - 30)$ Factor each group: $x^2(2x + 3) - 10(2x + 3)$ Factor out the common binomial factor: $(2x + 3)(x^2 - 10)$

3. $x^3 - 3x^2 - x + 3$

Again, we use grouping: $(x^3 - 3x^2) + (-x + 3)$ Factor each group: $x^2(x - 3) - 1(x - 3)$ Factor out the common binomial factor: $(x - 3)(x^2 - 1)$ The second factor can be factored further as: $(x - 3)(x - 1)(x + 1)$

4. $x^3 + 4x^2 - 7x + 2$

Using grouping: $(x^3 + 4x^2) + (-7x + 2)$ Factor each group: $x^2(x + 4) - 1(7x - 2)$ This polynomial does not factor easily using simple techniques, and will likely require advanced methods such as numerical factorization or the cubic formula.

5. $18x^3 - 57x^2 - 85x + 100$

We can attempt factoring by grouping again: $(18x^3 - 57x^2) + (-85x + 100)$ Factor each group: $3x^2(6x - 19) - 5(17x - 20)$ This expression doesn't easily factor further without using advanced techniques, so numerical or synthetic division methods might be better suited here.

Would you like to dive deeper into any specific factorization or proceed with solving the more complex cases?

Related Questions:

1. How do you factor polynomials using the grouping method?
2. What are some alternative methods for factoring cubic polynomials?
3. Can you always factor a polynomial over real numbers?
4. How does synthetic division help in factoring polynomials?
5. When is numerical factorization needed for polynomials?

Tip: Always check for the greatest common factor (GCF) before attempting to factor more complex expressions.