In order to factor this polynomial, first let's calculate the zeros using the quadratic formula:
[tex]\begin{gathered} x^2+13x+40=0\\ \\ a=1,b=13,c=40\\ \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\ \\ x=\frac{-13\pm\sqrt{169-160}}{2}\\ \\ x_1=\frac{-13+3}{2}=-5\\ \\ x_2=\frac{-13-3}{2}=-8 \end{gathered}[/tex]Now, we can write the polynomial in the factored form below:
[tex]\begin{gathered} a(x-x_1)(x-x_2)\\ \\ =(x+5)(x+8) \end{gathered}[/tex][tex]\begin{gathered} a(x-x_1)(x-x_2)\\ \\ =(x+5)(x+8) \end{gathered}[/tex]