Let's simplify each of the given expressions step by step.
### Expression 1: [tex]\(\frac{x^2 + 6}{4x - 16}\)[/tex]
1. Factor the Denominator:
The denominator can be factored as:
[tex]\[
4x - 16 = 4(x - 4)
\][/tex]
2. Rewrite the Expression:
Substitute the factored form of the denominator:
[tex]\[
\frac{x^2 + 6}{4(x - 4)}
\][/tex]
### Expression 2: [tex]\(\frac{5x^2}{x^2 - 16}\)[/tex]
1. Factor the Denominator:
The denominator can be factored as:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
2. Rewrite the Expression:
Substitute the factored form of the denominator:
[tex]\[
\frac{5x^2}{(x - 4)(x + 4)}
\][/tex]
### Expression 3: [tex]\(\frac{2x - 7}{2x + 8}\)[/tex]
1. Simplify the Numerator and Denominator:
There is no immediate factoring or simplification that can be done on each part, but we can simplify the coefficients.
2. Rewrite the Expression to Match the Simplified Coefficients:
Notice that:
[tex]\[
2(x - \frac{7}{2}) = 2x - 7 \quad \text{and} \quad 2(x + 4) = 2x + 8
\][/tex]
Therefore, the expression becomes:
[tex]\[
\frac{x - \frac{7}{2}}{x + 4}
\][/tex]
Putting it all together, the simplified forms of the given expressions are:
[tex]\[
\frac{x^2 + 6}{4(x - 4)}, \quad \frac{5x^2}{(x - 4)(x + 4)}, \quad \frac{x - \frac{7}{2}}{x + 4}
\][/tex]
