To determine which expression is equivalent to the given expression [tex]\(\frac{4m^2 - 9}{2m^2 - 13m + 15}\)[/tex], we need to simplify the provided fraction step by step.
1. Factor the numerator:
The numerator is [tex]\(4m^2 - 9\)[/tex]. This expression is a difference of squares, which can be factored as follows:
[tex]\[
4m^2 - 9 = (2m)^2 - 3^2 = (2m - 3)(2m + 3)
\][/tex]
2. Factor the denominator:
The denominator is [tex]\(2m^2 - 13m + 15\)[/tex]. We factor this quadratic expression by finding two numbers that multiply to [tex]\(2 \times 15 = 30\)[/tex] and add up to [tex]\(-13\)[/tex]. These numbers are [tex]\(-3\)[/tex] and [tex]\(-10\)[/tex]. Using these numbers, we can rewrite and factor the denominator:
[tex]\[
2m^2 - 13m + 15 = 2m^2 - 3m - 10m + 15 = m(2m - 3) - 5(2m - 3) = (m - 5)(2m - 3)
\][/tex]
3. Simplify the fraction:
Having factored both the numerator and the denominator, we replace the original expression with its factored form:
[tex]\[
\frac{4m^2 - 9}{2m^2 - 13m + 15} = \frac{(2m - 3)(2m + 3)}{(m - 5)(2m - 3)}
\][/tex]
Notice that the [tex]\((2m - 3)\)[/tex] term appears in both the numerator and the denominator, allowing us to cancel it:
[tex]\[
\frac{(2m - 3)(2m + 3)}{(m - 5)(2m - 3)} = \frac{2m + 3}{m - 5}
\][/tex]
Therefore, the expression equivalent to [tex]\(\frac{4m^2 - 9}{2m^2 - 13m + 15}\)[/tex] is [tex]\(\frac{2m + 3}{m - 5}\)[/tex].
Among the given choices, the correct answer is:
[tex]\(\frac{2m + 3}{m - 5}\)[/tex]
