I started sloving an algebric problem and i wonder if we can write $x^2+y^2+z^2$ or $a^2+b^2+c^2+2a+2b+2c+3$ as a product of terms.
By product of terms i think of writing does terms as a product:
Ex: $2a^2-3a-5=2a^2-5a+2a-5=(a+1)(2a-5)$
If yes, tell me so i can continue my problem. Hope one of you can help me! Thnk you.
You can if you are ok with complex numbers. For example, $$x^2+y^2+z^2 = \left(\sqrt{x^2+y^2}+iz\right)\left(\sqrt{x^2+y^2}-iz\right).$$
For the second problem: $$a^2+b^2+c^2+2a+2b+2c+3$$ $$=a(a+2+1/a)+b(b+2+1/b)+c(c+2+1/c)$$ $$=a(\sqrt{a}+\frac{1}{\sqrt{a}})^2+b(\sqrt{b}+\frac{1}{\sqrt{b}})^2+c(\sqrt{c}+\frac{1}{\sqrt{c}})^2$$ $$=(a+1)^2+(b+1)^2+(c+1)^2.$$ Then, apply above factorisation of $x^2+y^2+z^2$. This is at least a partial factoring.