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To factor the expression \(h^3 - 27\), we identify it as a difference of cubes. A difference of cubes occurs when you have an expression in the form \(a^3 - b^3\).
Here, \(h^3 - 27\) fits this form with \(a = h\) and \(b = 3\) because \(27 = 3^3\). Once identified, we can use the difference of cubes formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] By substituting our values into the formula, we get: \[ h^3 - 27 = (h - 3)(h^2 + 3h + 9) \] This formula helps simplify and factor expressions that are originally unwieldy. It’s always crucial to recognize the cube terms first.

Factoring polynomials doesn't end once you have applied the formulas. You also need to check if the resulting factors can be broken down further.
After factoring \(h^3 - 27\) into \((h - 3)(h^2 + 3h + 9)\), the next step is to examine each factor individually.
The linear polynomial \(h - 3\) is already fully factored and cannot be simplified further.
Now, we examine \(h^2 + 3h + 9\). To check if it can be factored further, we look for real or complex roots.
If no further real or simple factors are found, the polynomial \(h^2 + 3h + 9\) is considered a prime polynomial. Here, we discover that it does not factor over the reals, confirming it as prime. This step ensures complete accuracy in your final factorization.

Various factoring techniques help in simplifying polynomials, and they are vital tools in algebra. For this exercise, we used the difference of cubes formula.
There are multiple other techniques, such as:

• Factoring out the greatest common factor (GCF)
• Factoring by grouping
• Factoring trinomials
• Using special formulas like the sum of cubes or perfect square trinomials

These techniques can be applied based on the polynomial's structure.
It's also essential to recheck simplified forms using polynomial prime checks, ensuring no overlooked factors. Practicing various factoring methods builds proficiency in handling diverse polynomial expressions efficiently. Always try to identify the polynomial form first, such as recognizing a binomial, trinomial, or a special form like a difference of squares or cubes. Then apply the most suitable technique for accurate factorization.