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Factorise x45x2+4<0

Answer
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Hint: Firstly try to simplify the given higher degree equation into a smaller degree equation. Then depending upon the degree of the equation, try to factorize it into as many factors as possible. Based on the factors obtained, deduce the interval in which the value of the given variable lies.

Complete step-by-step answer:
We are given with the inequality x45x2+4<0
This can be rewritten as (x2)25x2+4<0
This becomes a quadratic equation in terms of x2.
We know that any quadratic equation in the variable x is of the form ax2+bx+c=0 .
Therefore using the quadratic formula to find the values of roots we get
x=b±b24ac2a
Here in this question we have the inequality x45x2+4<0 which is a quadratic in terms of x2.
Therefore using the quadratic formula to find the values of roots we get
x2=b±b24ac2a
Here a=1,b=5,c=4
Hence we get x2=(5)±(5)24(1)(4)2(1)
On doing the calculations in the root part we get
x2=5±25162
x2=5±32
Therefore we get x2=4,1
Hence the inequality x45x2+4<0 can be written as (x21)(x24)<0
Using the identity (a+b)(ab)=a2b2 we get
So, the correct answer is “(x1)(x+1)(x2)(x+2)<0”.

Note: Factorization or factoring is defined as the breaking or decomposition of an entity (for example a number, a matrix, or a polynomial) into a product of another entity, or factors, which when multiplied together gives the original number. In the factorization method, we reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of factors instead of expanding the brackets.
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