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Let's work through the factoring process for each of the given functions.

### a) \( f(x) = x^3 + 9x^2 + 8x - 60 \)

Step 1: Try grouping terms.
- Group the first two terms and the last two terms:  
  \( (x^3 + 9x^2) + (8x - 60) \)

Step 2: Factor out the greatest common factor (GCF) from each group.
- From \( x^3 + 9x^2 \), factor out \( x^2 \):  
  \( x^2(x + 9) \)
- From \( 8x - 60 \), factor out 4:  
  \( 4(2x - 15) \)

The expression is now:  
\( x^2(x + 9) + 4(2x - 15) \)

This does not factor further easily by grouping, so this is likely the final factored form for now.

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### b) \( f(x) = x^3 - 7x - 6 \)

Step 1: Use synthetic or trial and error for factoring. Try factoring using possible roots of the cubic equation.

Possible rational roots are factors of -6 (constant term) divided by factors of 1 (leading coefficient of \(x^3\)), i.e., \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Testing \( x = -1 \):
- Substituting into the function:  
  \( (-1)^3 - 7(-1) - 6 = -1 + 7 - 6 = 0 \)

So, \( x = -1 \) is a root.

Step 2: Use synthetic division to divide \( f(x) \) by \( x + 1 \).

Performing the division of \( x^3 - 7x - 6 \) by \( x + 1 \) gives \( x^2 - x - 6 \).

Step 3: Factor \( x^2 - x - 6 \).
- \( x^2 - x - 6 = (x - 3)(x + 2) \)

Thus, the fully factored form is:  
\( f(x) = (x + 1)(x - 3)(x + 2) \)

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### c) \( f(x) = x^4 - 5x^2 + 4 \)

Step 1: Let \( y = x^2 \), so the equation becomes:  
\( y^2 - 5y + 4 \)

Step 2: Factor the quadratic in \( y \).  
\( y^2 - 5y + 4 = (y - 4)(y - 1) \)

Step 3: Substitute back \( y = x^2 \).
- \( (x^2 - 4)(x^2 - 1) \)

Step 4: Factor the difference of squares.  
- \( x^2 - 4 = (x - 2)(x + 2) \)  
- \( x^2 - 1 = (x - 1)(x + 1) \)

Thus, the fully factored form is:  
\( f(x) = (x - 2)(x + 2)(x - 1)(x + 1) \)

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Would you like any further details on these steps or any clarifications?

### Additional Practice Questions:
1. How do you apply synthetic division to cubic functions?
2. What are the common techniques for factoring polynomials by grouping?
3. Can you explain the process of factoring the difference of squares?
4. How do you determine possible rational roots of a polynomial?
5. What are the techniques to factor higher-degree polynomials, such as quartic ones?

### Tip:
When factoring polynomials, always check for possible common factors and apply grouping or synthetic division techniques for higher-degree polynomials.

Factor the following functions fully:
a) f(x) = x^3 + 9x^2 + 8x - 60
b) f(x) = x^3 - 7x - 6
c) f(x) = x^4 - 5x^2 + 4

Learn how to fully factor cubic and quartic polynomial functions, including grouping, synthetic division, and the difference of squares. Problems include f(x) = x^3 + 9x^2 + 8x - 60 and f(x) = x^4 - 5x^2 + 4. Suitable for high school students in Grades 9-12.

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