1. **State the problem:** Factor the expression $$9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc$$. 2. **Recognize the form:** This is a quadratic expression in three variables $a$, $b$, and $c$ with mixed terms. 3. **Try to express as a perfect square of a trinomial:** Assume it factors as $$(x a + y b + z c)^2$$ for some constants $x$, $y$, and $z$. 4. **Expand the assumed form:** $$ (x a + y b + z c)^2 = x^2 a^2 + y^2 b^2 + z^2 c^2 + 2 x y a b + 2 x z a c + 2 y z b c $$ 5. **Match coefficients with the original expression:** - Coefficient of $a^2$: $x^2 = 9$ so $x = 3$ or $-3$. - Coefficient of $b^2$: $y^2 = 1$ so $y = 1$ or $-1$. - Coefficient of $c^2$: $z^2 = 4$ so $z = 2$ or $-2$. - Coefficient of $ab$: $2 x y = -6$ so $x y = -3$. - Coefficient of $ac$: $2 x z = 12$ so $x z = 6$. - Coefficient of $bc$: $2 y z = -4$ so $y z = -2$. 6. **Find consistent signs:** - From $x^2=9$, $x=3$ or $-3$. - From $x y = -3$, if $x=3$, then $y = -1$. - From $x z = 6$, if $x=3$, then $z=2$. - From $y z = -2$, with $y=-1$ and $z=2$, $y z = -2$ matches. 7. **Therefore, the factorization is:** $$ (3a - b + 2c)^2 $$ **Final answer:** $$9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc = (3a - b + 2c)^2$$