To solve the problem, we first need to clarify the function \( f(x) \) and its relationship to the given expression \( 4f(x) = x^5 + 5x^4 + 4 \). We'll first find the function \( f(x) \).
Starting from the equation: \[ 4f(x) = x^5 + 5x^4 + 4 \] we can divide both sides by 4 to express \( f(x) \): \[ f(x) = \frac{1}{4}(x^5 + 5x^4 + 4) \]
Now, we need to find the expression for \( f(x) \) explicitly: \[ f(x) = \frac{1}{4} x^5 + \frac{5}{4} x^4 + 1 \]
Next, we need to compute \( f(x)f(x) \). This means we will compute \( f(x) \times f(x) \). Let's denote \( f(x) \) simply as: \[ f(x) = \frac{1}{4} x^5 + \frac{5}{4} x^4 + 1 \]
Now, we need \( f(x)f(x) \): \[ f(x)f(x) = \left(\frac{1}{4} x^5 + \frac{5}{4} x^4 + 1\right) \left(\frac{1}{4} x^5 + \frac{5}{4} x^4 + 1\right) \]
Now, we will simplify this expression. We can expand it using distributive property (i.e., the FOIL method) to find out \( f(x)f(x) \): \[ f(x)f(x) = \left(\frac{1}{4} x^5\right)^2 + 2\left(\frac{1}{4} x^5\right)\left(\frac{5}{4} x^4\right) + 2\left(\frac{1}{4} x^5\right)(1) + \left(\frac{5}{4} x^4\right)^2 + 2\left(\frac{5}{4} x^4\right)(1) + 1^2 \] Calculating the terms:
- \( \left(\frac{1}{4} x^5\right)^2 = \frac{1}{16} x^{10} \)
- \( 2\left(\frac{1}{4} x^5\right)\left(\frac{5}{4} x^4\right) = \frac{10}{16} x^9 = \frac{5}{8} x^9 \)
- \( 2\left(\frac{1}{4} x^5\right)(1) = \frac{2}{4} x^5 = \frac{1}{2} x^5 \)
- \( \left(\frac{5}{4} x^4\right)^2 = \frac{25}{16} x^8 \)
- \( 2\left(\frac{5}{4} x^4\right)(1) = \frac{10}{4}x^4 = \frac{5}{2} x^4 \)
- \( 1^2 = 1 \)
Combining these gives: \[ f(x)f(x) = \frac{1}{16} x^{10} + \frac{5}{8} x^9 + \frac{25}{16} x^8 + \frac{1}{2} x^5 + \frac{5}{2} x^4 + 1 \]
Next, we need to find the remainder when \( f(x)f(x) \) is divided by \( x + 3 \). We can use the Remainder Theorem, which states that the remainder of \( p(x) \) when divided by \( x-c \) is \( p(c) \).
Let’s evaluate \( f(x)f(x) \) at \( x = -3 \): \[ f(-3) = \frac{1}{4}(-3)^5 + \frac{5}{4}(-3)^4 + 1 \] Calculating \( (-3)^5 = -243 \) and \( (-3)^4 = 81 \): \[ f(-3) = \frac{1}{4}(-243) + \frac{5}{4}(81) + 1 = -\frac{243}{4} + \frac{405}{4} + 1 = \frac{162}{4} + 1 = \frac{162}{4} + \frac{4}{4} = \frac{166}{4} = \frac{83}{2} \] Thus, \[ f(-3)f(-3) = \left(\frac{83}{2}\right)^2 = \frac{6889}{4} \]
Finally, the remainder when \( f(x)f(x) \) is divided by \( x + 3 \) is: \[ \frac{6889}{4} \]
Therefore, the remainder is: \[ \boxed{\frac{6889}{4}} \]