We are given two functions, $f(x) = 2x^2 - 5x$ and $g(x) = 5x + 4$. We need to find the following composite functions: $f(g(x))$, $g(f(x))$, $f(f(x))$, and $g(g(x))$.

1. Problem Description

We are given two functions, $f(x) = 2x^2 - 5x$ and $g(x) = 5x + 4$. We need to find the following composite functions: $f(g(x))$, $g(f(x))$, $f(f(x))$, and $g(g(x))$.

2. Solution Steps

(a) $f(g(x))$

We need to substitute $g(x)$ into $f(x)$.

$f(g(x)) = 2(g(x))^2 - 5(g(x))$

$f(g(x)) = 2(5x+4)^2 - 5(5x+4)$

$f(g(x)) = 2(25x^2 + 40x + 16) - 25x - 20$

$f(g(x)) = 50x^2 + 80x + 32 - 25x - 20$

$f(g(x)) = 50x^2 + 55x + 12$

(b) $g(f(x))$

We need to substitute $f(x)$ into $g(x)$.

$g(f(x)) = 5(f(x)) + 4$

$g(f(x)) = 5(2x^2 - 5x) + 4$

$g(f(x)) = 10x^2 - 25x + 4$

(c) $f(f(x))$

We need to substitute $f(x)$ into $f(x)$.

$f(f(x)) = 2(f(x))^2 - 5(f(x))$

$f(f(x)) = 2(2x^2 - 5x)^2 - 5(2x^2 - 5x)$

$f(f(x)) = 2(4x^4 - 20x^3 + 25x^2) - 10x^2 + 25x$

$f(f(x)) = 8x^4 - 40x^3 + 50x^2 - 10x^2 + 25x$

$f(f(x)) = 8x^4 - 40x^3 + 40x^2 + 25x$

(d) $g(g(x))$

We need to substitute $g(x)$ into $g(x)$.

$g(g(x)) = 5(g(x)) + 4$

$g(g(x)) = 5(5x+4) + 4$

$g(g(x)) = 25x + 20 + 4$

$g(g(x)) = 25x + 24$

3. Final Answer

(a) $f(g(x)) = 50x^2 + 55x + 12$

(b) $g(f(x)) = 10x^2 - 25x + 4$

(c) $f(f(x)) = 8x^4 - 40x^3 + 40x^2 + 25x$

(d) $g(g(x)) = 25x + 24$