The equation given in the image is:

$3\cos^2(x) + \cos(x) - 4$

Steps to Factor and Simplify:

1. Substitute $u = \cos(x)$: This simplifies the equation into a quadratic form: $3u^2 + u - 4$

2. Factor the quadratic equation: We need to find factors of $3u^2 + u - 4$. We look for two numbers that multiply to $3 \times -4 = -12$ and add to $1$. The pair $4$ and $-3$ works.

   Now split the middle term: $3u^2 + 4u - 3u - 4$ Group terms: $(3u^2 - 3u) + (4u - 4)$ Factor each group: $3u(u - 1) + 4(u - 1)$ Factor out the common binomial $(u - 1)$: $(3u + 4)(u - 1)$

3. Substitute back $u = \cos(x)$: $(3\cos(x) + 4)(\cos(x) - 1)$

Thus, the factored form of the expression is: $(3\cos(x) + 4)(\cos(x) - 1)$

Final Simplified Answer:

$(3\cos(x) + 4)(\cos(x) - 1)$

Would you like more details on any part of the process or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you solve a quadratic equation using the factoring method?
2. What are some other trigonometric identities that can be factored like this?
3. How can this factoring method be applied to solve for $x$ in trigonometric equations?
4. What are alternative methods to solving trigonometric quadratics besides factoring?
5. How does the substitution method simplify complex trigonometric expressions?

Tip: When dealing with quadratic trigonometric expressions, always check if substitution can simplify the equation into a solvable form!