<s>[INST] Write the quartic equation $4x^4+12x^3-35x^2-300x+625$ as the product of two quadratic expressions. Write the quartic expression 
$4x^4+12x^3-35x^2-300x+625$ 
as the product of two quadratic expressions with real coefficients.
I would like to know the way of solving it other than solving with the simultaneous equation. 
 [/INST] By the Ruffini's theorem, I have that $x=\frac{5}{2}$ is a rational zero of the polynomial. So, dividing by $x-\frac{5}{2}$, I have: $P(x)=(x-\frac{5}{2})(4x^3+22x^2+20x-250)=(x-\frac{5}{2})$. Using another time Ruffini's theorem, I have $\frac{5}{2}$ a zero, so $P(x)$ becomes: $$P(x)=2(x-\frac{5}{2})^2(x^2+8x+25)$$ $x^2+8x+25$ is irriducibile in $R$, so the factorization stop.
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