If $a(x+b)=3 x+12$ for all values of $x$, then $a+b$ equals
(A) 12
(B) 15
(C) 8
(D) 7
(E) 13
Solution 1

Since $a(x+b)=3 x+12$ for all $x$, then $a x+a b=3 x+12$ for all $x$.

Since the equation is true for all $x$, then the coefficients on the left side must match the coefficients on the right side.

Therefore, $a=3$ and $a b=12$, which gives $3 b=12$ or $b=4$.

Finally, $a+b=3+4=7$.

## Solution 2

Since $a(x+b)=3 x+12$ for all $x$, then the equation is true for $x=0$ and $x=1$.

When $x=0$, we obtain $a(0+b)=3(0)+12$ or $a b=12$.

When $x=1$, we obtain $a(1+b)=3(1)+12$ or $a+a b=15$.

Since $a b=12$, then $a+12=15$ or $a=3$.

Since $a b=12$ and $a=3$, then $b=4$.

Finally, $a+b=3+4=7$.

ANSWER: (D)

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