To convert the fraction [tex]\(\frac{w-3}{w+5}\)[/tex] to an equivalent fraction with the denominator [tex]\(w^2 + w - 20\)[/tex], we need to ensure that the numerators and denominators are adjusted properly so that the values of the fractions remain unchanged.
1. Identify the Original Fraction:
The given fraction is [tex]\(\frac{w-3}{w+5}\)[/tex].
2. Factor the Target Denominator:
The target denominator is [tex]\(w^2 + w - 20\)[/tex]. We need to see if this can be factored.
[tex]\[
w^2 + w - 20 = (w + 5)(w - 4)
\][/tex]
3. Rewriting the Target Denominator:
Given that the original fraction's denominator is [tex]\(w + 5\)[/tex], we recognize that:
[tex]\[
\frac{w-3}{w+5} = \frac{(w-3) \cdot \frac{(w-4)}{(w-4)}}{(w+5) \cdot \frac{(w-4)}{(w-4)}} = \frac{(w-3)(w-4)}{(w+5)(w-4)}
\][/tex]
4. Multiply the Numerator and Denominator:
Multiply the numerator and the denominator by [tex]\((w - 4)\)[/tex] to obtain the equivalent fraction with the new denominator [tex]\(w^2 + w - 20\)[/tex]:
[tex]\[
\frac{(w - 3)(w - 4)}{(w + 5)(w - 4)} = \frac{(w - 3)(w - 4)}{w^2 + w - 20}
\][/tex]
5. Expand the Numerator:
Expand [tex]\((w - 3)(w - 4)\)[/tex]:
[tex]\[
(w - 3)(w - 4) = w(w - 4) - 3(w - 4) = w^2 - 4w - 3w + 12 = w^2 - 7w + 12
\][/tex]
6. Equivalent Fraction:
Therefore, the equivalent fraction with the denominator [tex]\(w^2 + w - 20\)[/tex] is:
[tex]\(
\frac{w^2 - 7w + 12}{w^2 + w - 20}
\)[/tex]
Thus, the correct answer is:
D. [tex]\(\frac{w^2 - 7w + 12}{w^2 + w - 20}\)[/tex]
