To solve the given problem, we first multiply the two rational expressions:
[tex]\[
\frac{3 x^2+2 x-21}{-2 x^2-2 x+12} \cdot \frac{2 x^2+25 x+63}{6 x^2+7 x-49}
\][/tex]
Upon simplifying the product, we want to express it in the form:
[tex]\[
\frac{a x + b}{c x + d}
\][/tex]
where [tex]\( a = 1 \)[/tex]. We need to find the values of [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] that satisfy this equivalence.
After performing the necessary calculations, we find that:
[tex]\[
b = 9
\][/tex]
[tex]\[
c = -2
\][/tex]
[tex]\[
d = 4
\][/tex]
Thus, the values are:
[tex]\[
b = 9, \quad c = -2, \quad d = 4
\][/tex]
