Problem 401

Since $(x+a)(x+8)=x^{2}+bx+24$ for all $x$, then $x^{2}+ax+8x+8a=x^{2}+bx+24$ or $x^{2}+(a+8)x+8a=x^{2}+bx+24$ for all $x$. Since the equation is true for all $x$, then the coefficients on the left side must match the coefficients on the right side. Therefore, $a+8=b$ and $8a=24$. The second equation gives $a=3$, from which the first equation gives $b=3+8=11$. Finally, $a+b=3+11=14$.