Let $f(x)=x+5$ and let $g(x)=x^2+1$. Let $p(x)=g(x)+f(x)$ and let $q(x)=g(x)-f(x)$. Find $p(x)\cdot q(x)$.

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

+25 = x

step-by-step explanation:

here x is the hypotenuse of a right triangle whose legs are 7 and 24.   by the pythagorean theorem, 7² + 24² must equal x²:

x² = 49 + 576 = 625.   thus, x = +√625 = +25 = x

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step-by-step explanation: