Final answer:
The cubic equation m³ + 4m - 21 is factored completely by finding two numbers that multiply to -21 and add up to 4, leading to the factored form (m - 3)(m² + 7).
Step-by-step explanation:
To factor the cubic equation m³ + 4m - 21 completely, you need to find two numbers that multiply to give -21 (the constant term) and add to give 4 (the coefficient of the middle term). After testing various factors of -21, we discover that 7 and -3 meet the criteria, since 7*(-3) = -21 and 7+(-3) = 4.
Now, we rewrite the middle term using the numbers 7 and -3:
m³ + 7m - 3m - 21
We can group them to factor by grouping:
m(m² + 7) - 3(m² + 7)
Notice that (m² + 7) is a common factor:
(m - 3)(m² + 7)
Therefore, the factored form of the equation is (m - 3)(m² + 7).