AP DSC School Assistant Math Model Test 1

Concept:The given polynomial can be rewritten by grouping terms to form a perfect square and then factoring as a quadratic in $(2x - y)$.Explanation:Step 1: Rearrange the polynomial:$$4x^{2} + y^{2} + 14x - 7y - 4xy + 12 = 4x^{2} + y^{2} - 4xy + 14x - 7y + 12$$Notice that $4x^{2} + y^{2} - 4xy = (2x - y)^{2}$.Step 2: Write the polynomial as:$$(2x - y)^{2} + 7(2x - y) + 12$$Let $a = 2x - y$. Then the expression becomes:$$a^{2} + 7a + 12$$Step 3: Factor the quadratic:$$a^{2} + 7a + 12 = a^{2} + 4a + 3a + 12 = (a + 4)(a + 3)$$Step 4: Substitute back $a = 2x - y$:$$(2x - y + 4)(2x - y + 3)$$Thus, the two factors are $2x - y + 4$ and $2x - y + 3$. Among the given options, $2x - y + 3$ is present.Answer:$2x - y + 3$ (Option C)