Simplify: \( \frac{x^{4}+x}{x^{3}-x^{2}+x} \) Hence: 2.2.1 Determine the value of \( \frac{x^{4}+x}{x^{3}-x^{2}+x} \), if \( x=-3 \) 2.2.2 solve for \( x \) if: \[ \frac{x^{4}+x}{x^{3}-x^{2}+x}=2 \]

To simplify the expression \( \frac{x^{4}+x}{x^{3}-x^{2}+x} \), we start by factoring both the numerator and the denominator. In the numerator: \[ x^{4} + x = x(x^{3} + 1) = x(x + 1)(x^{2}-x+1) \] In the denominator: \[ x^{3} - x^{2} + x = x(x^{2} - x + 1) \] Now we can rewrite the expression: \[ \frac{x(x + 1)(x^{2}-x+1)}{x(x^{2}-x+1)} \] Assuming \( x \neq 0 \) and \( x^{2}-x+1 \neq 0 \), we can cancel \( x \) and \( (x^{2}-x+1) \): \[ \frac{x + 1}{1} = x + 1 \] Now, let's solve the subproblems: ### 2.2.1 To determine the value of \( \frac{x^{4}+x}{x^{3}-x^{2}+x} \) at \( x = -3 \): Substituting \( x = -3 \) into the simplified expression \( x + 1 \): \[ -3 + 1 = -2 \] So, \( \frac{x^{4}+x}{x^{3}-x^{2}+x} = -2 \) when \( x = -3 \). ### 2.2.2 To solve for \( x \) in the equation \( \frac{x^{4}+x}{x^{3}-x^{2}+x}=2 \): Using the simplified version \( x + 1 = 2 \): \[ x + 1 = 2 \] Subtracting 1 from both sides gives: \[ x = 1 \] Thus, the solutions are: \( \frac{x^{4}+x}{x^{3}-x^{2}+x} = -2 \) at \( x = -3 \), and \( x = 1 \) satisfies the equation \( \frac{x^{4}+x}{x^{3}-x^{2}+x} = 2 \).