Question

# what polynomial can be added to x^{2}+5x+1 to get a sum of 4x^{2}-3

what polynomial can be added to $$\displaystyle{x}^{{{2}}}+{5}{x}+{1}$$ to get a sum of $$\displaystyle{4}{x}^{{{2}}}-{3}$$

When adding two polynomials, you need to add the like terms. Therefore, to figure out what polynomial can be added to $$\displaystyle{x}^{{{2}}}+{5}{x}+{1}{x}$$ to get a sum of $$\displaystyle{4}{x}^{{{2}}}−{3}$$, you can compare each pair of like terms to see what the missing like terms must be.
Looking at the quadratic terms, you need $$\displaystyle{x}^{{{2}}}+?={4}{x}^{{{2}}}$$. The quadratic term of the missing polynomial must then be 3x2 since $$\displaystyle{x}^{{{2}}}+{3}{x}^{{{2}}}={4}{x}^{{{2}}}$$
Looking at the linear terms, you need 5x+?=0 since the sum $$\displaystyle{4}{x}^{{{2}}}−{3}$$ does not have a linear term. The linear term of the missing polynomial must then be −5x since 5x+(−5x)=0.
Since the missing polynomial must have a quadratic term of $$\displaystyle{3}{x}^{{{2}}}$$, a linear term of −5x, and a constant term of −4, then the polynomial $$\displaystyle{3}{x}^{{{2}}}−{5}{x}−{4}$$ can be added to $$\displaystyle{x}^{{{2}}}+{5}{x}+{1}$$ to get a sum of $$\displaystyle{4}{x}^{{{2}}}−{3}$$