<s>[INST] Prove this has real roots $(a^2-bc)x^2+(a+b)(a-c)x+a(b-c)=0$ Prove this has real roots 
$(a^2-bc)x^2+(a+b)(a-c)x+a(b-c)=0$
My Work
\begin{align*}
\Delta&=(a+b)^2(a-c)^2-4a(b-c)(a^2-bc) \\
&=a^4+2a^3c+a^2c^2-2a^3b+b^2a^2-4a^2bc-2abc^2+2ab^2c+b^2c^2.
\end{align*}
How do I show that this is positive? Simplification doesn't help either...How to factor them? please help!
 [/INST] Hint: remarkably, that expression for $\Delta$ is a perfect square...! Can you factor the expression knowing that?
</s>