What polynomial added to $x^{2} + 4 x - 5$ gives $2 x^{2} - 3 x + 1$ ?

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Solution

Let the polynomial that should be added to the polynomial $x^{2} + 4 x - 5$ be $A$ , so that the resulting polynomial is $2 x^{2} - 3 x + 1$ . Therefore, we have:

$(x^{2} + 4 x - 5) + A = 2 x^{2} - 3 x + 1 \Rightarrow A = (2 x^{2} - 3 x + 1) - (x^{2} + 4 x - 5) \Rightarrow A = 2 x^{2} - 3 x + 1 - x^{2} - 4 x + 5 \Rightarrow A = (2 x^{2} - x^{2}) + (- 3 x - 4 x) + (1 + 5) \Rightarrow A = x^{2} - 7 x + 6$

Hence, $x^{2} - 7 x + 6$ should be added to the polynomial $x^{2} + 4 x - 5$ , so that the resulting polynomial is $2 x^{2} - 3 x + 1$ .

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