Certainly! Let's perform the operation and simplify the given expression step by step.
Step 1: Write down the given expression.
[tex]\[ \frac{x^2 + 10x + 24}{3x^2 + 3x} \div (x + 6) \][/tex]
Step 2: Factorize the numerator and denominator where possible.
- The quadratic [tex]\(x^2 + 10x + 24\)[/tex] can be factored as [tex]\((x + 4)(x + 6)\)[/tex], because [tex]\((x + 4)(x + 6) = x^2 + 10x + 24\)[/tex].
- The quadratic [tex]\(3x^2 + 3x\)[/tex] can be factored as [tex]\(3x(x + 1)\)[/tex], because [tex]\(3x(x + 1) = 3x^2 + 3x\)[/tex].
So, the given expression can be rewritten as:
[tex]\[ \frac{(x + 4)(x + 6)}{3x(x + 1)} \div (x + 6) \][/tex]
Step 3: Change the division to multiplication by the reciprocal of [tex]\((x + 6)\)[/tex].
[tex]\[ \frac{(x + 4)(x + 6)}{3x(x + 1)} \times \frac{1}{x + 6} \][/tex]
Step 4: Simplify the expression by canceling common factors in the numerator and the denominator.
- The [tex]\((x + 6)\)[/tex] in the numerator and denominator cancels out.
[tex]\[ \frac{(x + 4) \cancel{(x + 6)}}{3x(x + 1)} \times \frac{1}{\cancel{x + 6}} = \frac{x + 4}{3x(x + 1)} \][/tex]
The simplest form of the expression is:
[tex]\[ \frac{x + 4}{3x(x + 1)} \][/tex]
So, the final simplified form of the given expression is:
[tex]\[ \frac{x + 4}{3x(x + 1)} \][/tex]
