What is (2x^4+4x^3-5x^2+2x-3)/(2x^2+1) ?

The solution to (2x^4+4x^3-5x^2+2x-3)/(2x^2+1) is (x-1)(x+3)

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simplify (2x^4+4x^3-5x^2+2x-3)/(2x^2+1)

simplify

\frac{2 x ^{4} + 4 x ^{3} - 5 x ^{2} + 2 x - 3}{2 x ^{2} + 1}

(x - 1) (x + 3)

Solution steps

\frac{2 x ^{4} + 4 x ^{3} - 5 x ^{2} + 2 x - 3}{2 x ^{2} + 1}

Use the rational root theorem

= \frac{( x - 1 ) \frac{2 x ^{4} + 4 x ^{3} - 5 x ^{2} + 2 x - 3}{x - 1}}{2 x ^{2} + 1}

\frac{2 x ^{4} + 4 x ^{3} - 5 x ^{2} + 2 x - 3}{x - 1} = 2 x^{3} + 6 x^{2} + x + 3

= \frac{( x - 1 ) ( 2 x ^{3} + 6 x ^{2} + x + 3 )}{2 x ^{2} + 1}

Factor

2 x^{3} + 6 x^{2} + x + 3 : (x + 3) (2 x^{2} + 1)

= \frac{( x - 1 ) ( x + 3 ) ( 2 x ^{2} + 1 )}{2 x ^{2} + 1}

Cancel the common factor:

2 x^{2} + 1

= (x - 1) (x + 3)

- \frac{1}{3} \times (- 2)

20 + x - x^{2}

8 2^{3}

- 8 \times (- 5)

(5 x)^{- 3}

What is (2x^4+4x^3-5x^2+2x-3)/(2x^2+1) ?
The solution to (2x^4+4x^3-5x^2+2x-3)/(2x^2+1) is (x-1)(x+3)