Cho a thỏa mãn a² – 5a + 2 = 0. Tìm giá trị biểu thức: P = a⁵ – a⁴ – 18a³ + 9a² – 5a + 2017 + (a⁴ – 40a² + 4) : a².

P = a⁵ – a⁴ – 18a³ + 9a² – 5a + 2017 + (a⁴ – 40a² + 4) : a²

P = (a⁵ – 5a⁴ + 2a³) + (4a⁴ – 20a³ + 8a²) + (a² – 5a + 2) + 2015 + \(\frac{{{a^4} - 40{a^2} + 4}}{{{a^2}}}\)

P = a³(a² – 5a + 2) + 4a²(a² – 5a + 2) + (a² – 5a + 2) + 2015 + \(\frac{{{a^4} - 40{a^2} + 4}}{{{a^2}}}\)

P = 2015 +\(\frac{{{a^4} - 40{a^2} + 4}}{{{a^2}}}\)

Lại có: a² – 5a = –2 ⇒ (a² – 5a)² = 4

⇒ a⁴ – 10a³ + 25a² = 4

P = 2015 + \(\frac{{{a^4} - 40{a^2} + 4}}{{{a^2}}}\)

\( = \frac{{{a^4} + 1975{a^2} + 4}}{{{a^2}}} = \frac{{4 + 10a\left( {{a^2} - 5a + 2} \right) + 4\left( {{a^2} - 5a + 2} \right) + 1996{a^2} - 4}}{{{a^2}}} = 1996\)

Vậy P = 1996.