Without actual division, prove that 2x^4 - 5x^3 + 2x^2 - x + 2 is divisible by x^2 + 3x + 2.

2x⁴ - 5x³ + 2x² - x + 2=

2x²(x²-3x+2) +x³-2x² -x +2=

2x²(x²-3x+2) + x(x²-3x+2) +x² -3x +2=

(x²-3x+2)(2x²+x+1)

since x²-3x+2 is the factor of the given polynomial, we can state the polynomial is divisible by x²-3x+2

note: x²+3x+2 is not the factor of the given polynomial

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