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Express the algebraic expression in a simplified manner: 4a2b212abc+9c2 .

Answer
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Hint: The given algebraic expression is a complicated expression generated by a simple expression that is binomial expansion of (xy)2 . We will use the formula of this expression and try to simplify the given expression in the shortest form possible.
Formula used:
The binomial expansion for the expression (xy)2 is given by
 (xy)2=x22xy+y2

Complete step-by-step answer:
The given algebraic expression is:
 4a2b212abc+9c2
The given equation looks similar to the expression x22xy+y2
Comparing the given equation to this expression we get:
 x2=4a2b2,2xy=12abc,y2=9c2
On solving each thing individually we have:
 x2=4a2b2x=4a2b2=(2ab)2=2ab --(1)
 2xy=12abc=2×(2ab)×(3c) --(2)
 y2=9c2y=9c2=(3c)2=3c --(3)
Since, from the binomial expansion of (xy)2 we have:
x22xy+y2=(xy)2
Putting the values obtained in (1),(2) and (3) we get
 4a2b212ab+9c2=(2ab)22×(2ab)×(3c)+(3c)2
 4a2b212ab+9c2=(2ab3c)2
Therefore, the simplification of the algebraic expression 4a2b212abc+9c2 gives us (2ab3c)2 .
So, the correct answer is “ (2ab3c)2 ”.

Additional information:
There are various formulas one should remember to make simplification easier such as:
1) (xy)2=x22xy+y2
2) (x+y)2=x2+2xy+y2
3) (xy)(x+y)=x2y2
With these formulas in mind, all algebraic expressions can be simplified and expanded as well. These formulas can also be verified. These are called the special binomial formulas. This formula was given by Greek mathematician Euclid, later it was generalized to (xy)n . By simple multiplication we can prove that (x+y)2=x2+2xy+y2 and the other formulas too.

Note: Looking at the question first see all the possibilities to find the simplified answer. Think what formulas you are going to use. If you are not clear about the expression first try to split it into various terms, it will always give you an idea of factoring the expression.