Answer:
The data that we have is:
f(x) = x + 5
g(x) = x^2 + 1
p(x) = g(x) + f(x) = (x^2 + 1) + (x + 5) = x^2 + x + 6.
q(x) = g(x) - f(x) = (x^2 + 1) - (x + 5) = x^2 - x - 4
We want to find p(x)*q(x)
well, we can replace:
p(x)*g(x) = (g(x) + f(x))*(g(x) - f(x))
Now, you can recall the relationship:
a^2 - b^2 = (a + b)*(a - b)
then we have that:
(g(x) + f(x))*(g(x) - f(x)) = g(x)^2 - f(x)^2
now we can replace g(x) and f(x) by the expressions that we know:
g(x)^2 - f(x)^2 = (x^2 + 1)^2 - (x + 5)^2
now we can simplify this:
(x^2 + 1)^2 - (x + 5)^2 = (x^4 + 2*1*x + 1^2) - (x^2 + 2*5*x +5^2)
= x^4 + 2*x + 1 - x^2 - 10x - 25
= x^4 - x^2 - 8*x - 24
p(x)*q(x) = x^4 - x^2 - 8*x - 24
