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.

Looking at the constant terms, you need 1+?=−31+?=−3. The constant term of the missing polynomial must then be −4 since 1+(−4)=−3.

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 \(\displaystyle{4}{x}^{{{2}}}−{3}\)

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.

Looking at the constant terms, you need 1+?=−31+?=−3. The constant term of the missing polynomial must then be −4 since 1+(−4)=−3.

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 \(\displaystyle{4}{x}^{{{2}}}−{3}\)