Question 1116336
.
This problem looks strange.  It has only one unknown,  but imposes two conditions on it.

So,  formally,  it is over constrained.  Let's look into the solution.


<pre>
The fact that  x^3+2x^2+ax+6  gives the remainder of 4 when divided by x-1  means that  the value of the polynomial at x= 1 is equal to 4

(according to the Remainder theorem):


1^3 + 2*1^2 + a*1 + 6 = 4,   or

1 + 2 + a + 6 = 4,   which implies

a = 4 - 1 - 2 - 6 = -5.



The fact that  x^3+2x^2+ax+6  gives the remainder of 16 when divided by x+2  means that  the value of the polynomial at x= -2 is equal to 16

(according to the Remainder theorem):


(-2)^3 + 2*(-2)^2 + a*(-2) + 6 = 16,   or

-8 + 8 - 2a + 6 = 16,   which implies

-2a = 16 + 8 - 8 - 6 = 10  ====>  a = {{{10/(-2)}}} = -5.


Fortunately, both conditions give the same value for "a" equal to -5.


<U>Answer</U>.  a = -5.
</pre>


<U>Notice</U>.  The condition says "find &nbsp;a&nbsp; and &nbsp;b", &nbsp;&nbsp;but the polynomial has only &nbsp;"a" &nbsp;and has no &nbsp;"b".



==============


On the Remainder theorem see the lessons 


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Divisibility-of-polynomial-f%28x%29-by-binomial-x-a.lesson >Divisibility of polynomial f(x) by a binomial (x-a) and the Remainder theorem</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Solved-problems-on-the-Remainder-theorem.lesson>Solved problems on the Remainder theorem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Divisibility-of-a-polynomial-f%28x%29-by-the-binomial-x-a-and-the-Remainder-theorem.lesson>OVERVIEW of lessons on Divisibility of polynomial f(x) by binomial (x-a) and the Remainder theorem</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
"<U>Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem</U>".


Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.