Write the quartic equation $4x^4+12x^3-35x^2-300x+625$ as the product of two quadratic expressions.

From the rational root test we get $x=5/2$ as a double root.

Thus we have a factor of $(x-2.5)^2 = x^2-5x+6.25$

Upon dividing by $x^2-5x+6.25$ and bringing $4$ into the first factor we can factor it as

$$4x^4+12x^3-35x^2-300x+625=(4x^2-20x+25)(x^2+8x+25)$$

but obviously you do not have real roots for the second one.

I do not know if it is even possible to factor it as required

Tags:

Polynomials

Quartic Equations

Factoring

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