To solve the expression [tex]\(\frac{5 c^2 - d^2 + 3}{2 c - 4 d}\)[/tex], we need to evaluate both the numerator and the denominator separately and then simplify the fraction if possible.
### Step-by-step Solution:
#### Step 1: Evaluate the Numerator
The numerator is given by:
[tex]\[ 5 c^2 - d^2 + 3 \][/tex]
This is a straightforward expression involving the variables [tex]\(c\)[/tex] and [tex]\(d\)[/tex].
#### Step 2: Evaluate the Denominator
The denominator is given by:
[tex]\[ 2 c - 4 d \][/tex]
This is another straightforward expression involving the variables [tex]\(c\)[/tex] and [tex]\(d\)[/tex].
#### Step 3: Combine the Expressions
Now, combine the evaluated numerator and denominator into a fraction:
[tex]\[ \frac{5 c^2 - d^2 + 3}{2 c - 4 d} \][/tex]
#### Step 4: Simplify the Fraction
To simplify the fraction, look for common factors in the numerator and the denominator. However, in this case, [tex]\(5 c^2 - d^2 + 3\)[/tex] and [tex]\(2 c - 4 d\)[/tex] likely do not share common factors that can allow further simplification unless specific values of [tex]\(c\)[/tex] and [tex]\(d\)[/tex] satisfy them.
For a general form, the expression remains as it is:
[tex]\[ \frac{5 c^2 - d^2 + 3}{2 c - 4 d} \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ \frac{5 c^2 - d^2 + 3}{2 c - 4 d} \][/tex]
This is your final answer for the expression. If you have specific values for [tex]\(c\)[/tex] and [tex]\(d\)[/tex], you can plug them in to get a numerical result.
