Sure! Let's work through the problem step by step:
We are given two rational expressions:
[tex]\[ \frac{9x^2 + 3}{14x^2 - 9} \][/tex]
and
[tex]\[ \frac{-3x^2 + 11}{14x^2 - 9} \][/tex]
We need to subtract the second expression from the first. Since both denominators are the same, we can directly subtract the numerators:
[tex]\[ \frac{9x^2 + 3}{14x^2 - 9} - \frac{-3x^2 + 11}{14x^2 - 9} = \frac{(9x^2 + 3) - (-3x^2 + 11)}{14x^2 - 9} \][/tex]
Let's simplify the numerator:
[tex]\[ (9x^2 + 3) - (-3x^2 + 11) \][/tex]
Distribute the negative sign in the second part:
[tex]\[ 9x^2 + 3 + 3x^2 - 11 \][/tex]
Combine like terms:
[tex]\[ 9x^2 + 3x^2 + 3 - 11 = 12x^2 - 8 \][/tex]
So, the expression simplifies to:
[tex]\[ \frac{12x^2 - 8}{14x^2 - 9} \][/tex]
We can factor out a common factor from the numerator:
[tex]\[ 12x^2 - 8 = 4(3x^2 - 2) \][/tex]
Thus, we have:
[tex]\[ \frac{4(3x^2 - 2)}{14x^2 - 9} \][/tex]
So, the simplified form of the given subtraction is:
[tex]\[ \boxed{\frac{4(3x^2 - 2)}{14x^2 - 9}} \][/tex]
