To write the expression [tex]\( 4(3x + 2) + 2(x - 3) + 19 \)[/tex] in the form [tex]\( a(bx + c) \)[/tex] where [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] are integers and [tex]\( a > 1 \)[/tex], we will proceed through a detailed, step-by-step solution.
1. Expand the expression:
Start by expanding each term within the parentheses:
[tex]\[
4(3x + 2) = 4 \cdot 3x + 4 \cdot 2 = 12x + 8
\][/tex]
[tex]\[
2(x - 3) = 2 \cdot x - 2 \cdot 3 = 2x - 6
\][/tex]
2. Combine all terms:
Add the expanded terms together:
[tex]\[
12x + 8 + 2x - 6 + 19
\][/tex]
3. Simplify the expression:
Combine like terms to simplify the expression:
[tex]\[
12x + 2x + 8 - 6 + 19 = 14x + (8 - 6 + 19) = 14x + 21
\][/tex]
4. Factor out the greatest common divisor (GCD):
Combine the expression [tex]\( 14x + 21 \)[/tex] in the form [tex]\( a(bx + c) \)[/tex]. Notice that [tex]\( 14 \)[/tex] and [tex]\( 21 \)[/tex] have a common factor of [tex]\( 7 \)[/tex]:
[tex]\[
14x + 21 = 7(2x + 3)
\][/tex]
Thus, the expression [tex]\( 4(3x + 2) + 2(x - 3) + 19 \)[/tex] written in the form [tex]\( a(bx + c) \)[/tex], where [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] are integers and [tex]\( a > 1 \)[/tex], is:
[tex]\[
7(2x + 3)
\][/tex]
Here, [tex]\( a = 7 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 3 \)[/tex].
