Express \[ \frac{x^{4}}{3} \times \frac{6 x+24}{x^{5}+x^{4}} \] as a single fraction in its simplest form. Input Note: Enter answer in fully factorised form.
Solution
Step 1 ：First, let's simplify the given expression step by step. The given expression is: \[\frac{x^{4}}{3} \times \frac{6 x+24}{x^{5}+x^{4}}\]
Step 2 ：Factorise the terms in the expression. The term 6x + 24 can be factorised as 6(x + 4). The term x^5 + x^4 can be factorised as x^4(x + 1). So, the expression becomes: \[\frac{x^{4}}{3} \times \frac{6(x + 4)}{x^{4}(x + 1)}\]
Step 3 ：Simplify the expression. The x^4 in the numerator and denominator cancel out. So, the expression becomes: \[\frac{6(x + 4)}{3(x + 1)}\]
Step 4 ：Simplify further. The 6 and 3 in the numerator and denominator can be simplified to 2. So, the final simplified expression is: \[\frac{2(x + 4)}{x + 1}\]
Step 5 ：This is the simplest form of the given expression. Let's check if the result meets the requirements of the problem. The result is a single fraction in its simplest form, so it meets the requirements. The final answer is \[\boxed{\frac{2(x + 4)}{x + 1}}\]
