Answer: The required value of c that makes the given statement true is 5.
Step-by-step explanation: We are given to find the value of c that makes the following statement TRUE :
[tex]-2x^3(cx^3+x^2)=-10x^6-2x^5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
To find the value of c, we need to equate the coefficients of the same powers of the unknown variable x.
From equation (i), we have
[tex]-2x^3(cx^3+x^2)=-10x^6-2x^5\\\\\Rightarrow -2cx^{3+3}-2x^{3+2}=-10x^6-2x^5\\\\\Rightarrow -2cx^6-2x^5=-10x^6-2x^5.[/tex]
Equating the coefficients of [tex]x^6[/tex] on both sides of the above equation, we get
[tex]-2c=-10\\\\\Rightarrow c=\dfrac{-10}{-2}\\\\\Rightarrow c=5.[/tex]
Thus, the required value of c that makes the given statement true is 5.
