Question

If m(x) = x2 + 3 and n(x) = 5x + 9, which expression is equivalent to (mn)(x)?

Answer 1

For this case we have the following functions:
 $m (x) = x ^ 2 + 3 n (x) = 5x + 9$
 Multiplying both functions, we obtain an expression equivalent to (mn) (x).
 We have then:
 $(mn) (x) = m (x) n (x)$
 Substituting values we have:
 $(mn) (x) = (x ^ 2 + 3) (5x + 9)$
 Rewriting the expression we have:
 $(mn) (x) = (5x ^ 3 + 15x) + (9x ^ 2 + 27) (mn) (x) = 5x ^ 3 + 9x ^ 2 + 15x + 27$
 Answer:
 An expression that is equivalent to (mn) (x) is:
 $(mn) (x) = 5x ^ 3 + 9x ^ 2 + 15x + 27$

Answer 2

Solution:

It is given that,

$m(x)=x^2 +3, n(x)=5 x +9\\\\ mn(x)= (x^2+3)(5 x +9)\\\\ m n (x)=x^2\times (5 x +9)+3\times (5 x +9)\\\\ m n(x)=x^2 \times 5 x+x^2\times 9+3 \times 5 x +3 \times 9 \\\\m n(x)=5 x^3+9 x^2+15 x +27$

The equivalent expression to m n(x) is:

$1. x(5 x^2+9 x+15)+27\\\\ 2. 5 x^3+3 \times (3 x^2+5x +9) \\\\ 3.5 x^3 +3 x\times (3x +5)+27$

The Identity used here is

1. Distributive property of multiplication with respect to addition

 a × (b+c)= a × b + a × c

2. Law of indices

$a^m \times a^n=a^{m+n}$