The values of m for which m³ + 1 ≥ m(m + 1) are given by 

Concept:

(a2 - b2) = (a - b)(a + b)

Note: If the product of the two-term out of which one the term is always positive, is positive, the other term will also be positive.

Calculation:

Given that

m3 + 1 ≥ m(m + 1) 

⇒  m3 - m² - m + 1 ≥ 0

⇒ m²(m - 1) - 1(m - 1) ≥ 0

⇒ (m² - 1)(m - 1) ≥  0

∵ (a² - b²) = (a - b)(a + b)

⇒ (m - 1)²(m + 1) ≥ 0

We can see that, the term ( m - 1)² is either positive or 0 (at m = 1)

Hence, to make overall inequality positive, 

m + 1 ≥ 0

⇒ m ≥ -1

Hence, the value of m which satisfies given inequality is m ≥ -1.