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When adding or subtracting fractions, we need a common denominator. This is a shared value in the denominator of all the fractions. It allows us to combine them easily. In the problem given, the denominators are \(d+2\), \(d\), and \(d^2 + 2d\). To find a common denominator:

\[ d^2 + 2d \] can be factored as \(d(d+2)\). This factorization shows that \(d(d+2)\) is a common multiple of all denominators. Using this common denominator, \(d(d+2)\), we can rewrite each fraction:

• \(\frac{3d}{d+2}\)
• \(\frac{4}{d}\)
• \(\frac{d+8}{d^2 + 2d}\)

By converting each fraction to have the common denominator, we prepare for easy addition and subtraction.

Factoring is a crucial step in dealing with fractions, especially for simplifying and finding common denominators. Factoring breaks down complex expressions into products of simpler ones. For instance:

Consider the polynomial \(d^2 + 2d\). It can be factored by taking out the common factor \(d\), resulting in \(d(d+2)\). This step is vital because, in our given problem, by recognizing that \(d^2 + 2d = d(d+2)\), we identify a common denominator.

Here are a few pointers on factoring:

• Look for common factors in all terms.
• Use special factoring formulas, such as difference of squares.
• Break down higher-degree polynomials into products of first-degree (linear) factors when possible.

Factoring makes it possible to reduce expressions and find common denominators easily.

After combining fractions with a common denominator, the next step is to simplify the resulting expression.

Here, we combined the fractions: \[ \frac{3d^2}{d(d+2)} + \frac{4d + 8}{d(d+2)} - \frac{d+8}{d(d+2)} \]

Combine numerators: \[ \frac{3d^2 + (4d + 8) - (d+8)}{d(d+2)} = \frac{3d^2 + 4d + 8 - d - 8}{d(d+2)} \]
Combine like terms: \[ \frac{3d^2 + 3d}{d(d+2)} \]
Factor the numerator by extracting the greatest common factor (GCF) which is \(3d\). Then, we have: \( \frac{3d(d+1)}{d(d+2)} \).
Finally, cancel common factors in both numerator and denominator: \[ \frac{3(d+1)}{d+2} \]

Simplifying these fractions ensures the final result is in its simplest form. It makes the expression easier to understand and use.