\frac{4}{3x^2+15x}+\frac{7x}{x^2+10x+25}=

To simplify the expression 

\[
\frac{4}{3x^2 + 15x} + \frac{7x}{x^2 + 10x + 25},
\]

we first need to find a common denominator.

### Step 1: Factor the denominators
Let's start by factoring the denominators.

1. For the first fraction:
   \[
   3x^2 + 15x = 3x(x + 5)
   \]
   
2. For the second fraction:
   \[
   x^2 + 10x + 25 = (x + 5)(x + 5) = (x + 5)^2
   \]

So, the expression now becomes:
\[
\frac{4}{3x(x + 5)} + \frac{7x}{(x + 5)^2}
\]

### Step 2: Find the common denominator
The least common denominator (LCD) is the least common multiple of \(3x(x + 5)\) and \((x + 5)^2\). The LCD in this case is \(3x(x + 5)^2\).

### Step 3: Rewrite each fraction with the common denominator
1. The first fraction becomes:
   \[
   \frac{4}{3x(x + 5)} \times \frac{x + 5}{x + 5} = \frac{4(x + 5)}{3x(x + 5)^2}
   \]

2. The second fraction is already close, so multiply its denominator by \(3x\):
   \[
   \frac{7x}{(x + 5)^2} \times \frac{3x}{3x} = \frac{21x^2}{3x(x + 5)^2}
   \]

Now the expression becomes:
\[
\frac{4(x + 5)}{3x(x + 5)^2} + \frac{21x^2}{3x(x + 5)^2}
\]

### Step 4: Combine the fractions
Since both fractions now have the same denominator, we can combine them:

\[
\frac{4(x + 5) + 21x^2}{3x(x + 5)^2}
\]

### Step 5: Simplify the numerator
Expand and combine like terms in the numerator:

\[
4(x + 5) + 21x^2 = 4x + 20 + 21x^2
\]

So, the expression becomes:

\[
\frac{21x^2 + 4x + 20}{3x(x + 5)^2}
\]

This is the simplified form of the original expression.

Would you like to discuss this solution further or have any questions?

Here are 5 related questions to consider:
1. How can you factor and simplify the expression further if possible?
2. How do you determine the least common denominator for more complex fractions?
3. What are the steps to adding fractions with different denominators?
4. How would you solve for \(x\) if the simplified fraction was set equal to zero?
5. How do you handle fractions that involve polynomials in the numerator and denominator?

**Tip:** When dealing with complex fractions, always start by factoring the denominators to easily find the least common denominator.

Learn how to simplify the fractional expression 4/(3x^2 + 15x) + 7x/(x^2 + 10x + 25) step-by-step with a detailed solution. This problem involves algebraic manipulation, factoring, and finding common denominators. Suitable for high school students in Grades 10-12.

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