What are the solutions to x² + 10x + 21 = 0, found by completing the square?

Move the constant: $x^2+10x=-21$. Complete the square by adding $\left(\frac{10}{2}\right)^2=25$ to both sides: $x^2+10x+25=4$, so $(x+5)^2=4$. Then $x+5=\pm2$, giving $x=-3$ or $x=-7$, choice F. Check by factoring: $(x+3)(x+7)=x^2+10x+21$. Choice G ($3,7$) drops the negative signs. Choice H ($-3,7$) keeps only one sign. Choice J ($-21,1$) uses the factor pair of $21$ whose sum is $-20$ rather than $10$. The roots must sum to $-10$ and multiply to $21$. Either method gives the same roots, so use whichever is faster.