The polynomial which results from the expansion of $(x^2+5x+6)^2+(px+q)(x^3+7x^2+3x)$ has degree $2$. Find $p+q$.
(x^2+5x+6)^2+(px+q)(x^3+7x^2+3x)
x^4 + 10 x^3 + 37 x^2 + 60x + 36 + px^4 + 7px^3 + 3px^2 + qx^3 + 7qx^2 + 3qx
( 1 + p)x^4 + (10 + 7p + q) x^3 + (37 + 3p + 7q)x^2 +( 60+ 3q)x + 36
If the resulting polynomial is of degree 2, then
1 + p = 0 → p = -1
And
(10 + 7p + q) = 0
(10 - 7 + q) = 0
3 + q = 0
q = -3
So
(x^2+5x+6)^2+(-1x-3)(x^3+7x^2+3x) = 13 x^2 + 51 x + 36
So
p + q = ( -1 ) + (- 3) = - 4