How can we determine the factors of the expression x³ + 9x² + 5x + 45 using the method of grouping?

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To factor the polynomial expression x³ + 9x² + 5x + 45 by grouping, we can break it into two pairs of terms. Here’s a step-by-step guide:

  1. Group the terms: We first group the terms into two pairs:
    (x³ + 9x²) + (5x + 45)
  2. Factor out the common factors from each group:
    From the first group (x³ + 9x²), we can factor out x²:
    x²(x + 9)
    From the second group (5x + 45), we can factor out 5:
    5(x + 9)
  3. Write the factored expression:
    Now we can rewrite the original expression using these factors:
    x²(x + 9) + 5(x + 9)
  4. Factor out the common binomial:
    We see that (x + 9) is a common factor in both terms:
    (x + 9)(x² + 5)

Therefore, the factored form of the expression x³ + 9x² + 5x + 45 is:

(x + 9)(x² + 5)

This shows that we have successfully factored the polynomial by grouping, and we can confirm that this method is quite effective for certain types of polynomials.

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