To solve the equation
[tex]\[
\frac{x^2 + 5}{x + 5} = \frac{3}{x + 5}
\][/tex]
we follow these steps:
1. Identify Common Denominator:
Since both sides of the equation have the same denominator [tex]\(x + 5\)[/tex], we can equate the numerators directly (provided [tex]\(x \neq -5\)[/tex] to avoid division by zero).
2. Equate the Numerators:
Set the numerators equal to each other:
[tex]\[
x^2 + 5 = 3
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x^2 + 5 = 3
\][/tex]
Subtract 5 from both sides:
[tex]\[
x^2 = 3 - 5
\][/tex]
Simplify the right-hand side:
[tex]\[
x^2 = -2
\][/tex]
4. Consider the Result:
Taking the square root of both sides, we get:
[tex]\[
x = \sqrt{-2}
\][/tex]
The square root of a negative number is not a real number. Therefore, [tex]\(x = \sqrt{-2}\)[/tex] does not produce any real solutions.
Since the problem requires real solutions, we conclude that:
[tex]\[
\boxed{\text{There are no real solutions.}}
\][/tex]
