Find the square root of $4a^2 + b^2 + c^2 + 4ab - 2bc - 4ac$

### Concept & Formula This question requires factoring a quadratic polynomial with three variables. The underlying principle is the algebraic identity for the square of a trinomial: $$ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx $$ ### Step-by-Step Solution * The given expression under the root is $4a^2 + b^2 + c^2 + 4ab - 2bc - 4ac$. * Rewrite the squared terms to identify our base variables: $(2a)^2 + (b)^2 + (c)^2$. * Next, observe the signs of the cross-product terms: $+4ab$, $-2bc$, $-4ac$. * Because the terms involving $c$ are negative, but the term with just $a$ and $b$ is positive, the $c$ term must carry the negative sign in our factored form. * Let $x = 2a$, $y = b$, and $z = -c$. * Verify the cross terms against the formula: $$ 2xy = 2(2a)(b) = 4ab $$ $$ 2yz = 2(b)(-c) = -2bc $$ $$ 2zx = 2(2a)(-c) = -4ac $$ * The expression factors perfectly into the square: $(2a + b - c)^2$. * Taking the square root removes the square exponent, leaving $(2a + b - c)$. ### Exam Strategy & Shortcut Use the "Sign Check" method to bypass the rigorous algebra. Look exclusively at the mixed terms: $+4ab$, $-2bc$, $-4ac$. The negative signs are attached only to pairs containing the variable $c$, while the $ab$ pair is strictly positive. This logically dictates that $a$ and $b$ share the same sign, and $c$ has the opposite sign. Only the option $2a + b - c$ satisfies this exact rule. ### Common Pitfall The most common error is ignoring the negative signs entirely and carelessly assuming the answer is $(2a + b + c)$ because all the square terms are positive. Always trace the negative terms in the expanded polynomial back to identify which base variable carries the minus sign. ### Final Answer Therefore, the correct answer is **$2a + b - c$**.