How can a factoring problem be checked?

Hint: The factors that we obtain on solving algebraic equations can be checked for their true values. The factors that we obtain are also called the zeroes of the equation. So, we just have to substitute the factors in those equations and on solving the equation, we get the value of expression as 0, then the factor obtained is correct.

Complete step by step solution:
According to the given question, we are to provide a way to check whether the factor we obtained for an equation is correct or not.
In a quadratic equation, we generally try to solve the equation directly by factoring the equation and if it does not work that way, then we use the discriminant to find the values of the variable involved.
Let’s suppose, we were provided with a quadratic equation like \[{{x}^{2}}+2x+1\]. Let’s proceed to solve it, we get,
\[{{x}^{2}}+2x+1\]
\[\Rightarrow {{x}^{2}}+x+x+1\]
\[\Rightarrow x(x+1)+1(x+1)\]
\[\Rightarrow (x+1)(x+1)\]
Equating the factor to 0, we get the value as,
\[x=-1,-1\]
So, now we are to check if the value of \[x\] is correct or not.
So, we will substitute the values of \[x=-1\] in the given quadratic equation. If we get a value 0, we will know that the value of \[x\] is correct.
We have,
\[{{x}^{2}}+2x+1\]
Substituting \[x=-1\], we get,
\[\Rightarrow {{(-1)}^{2}}+2(-1)+1\]
\[\Rightarrow 1-2+1\]
\[\Rightarrow 2-2=0\]
Since, we got the value of the expression as 0, therefore, the value of \[x\] is correct.
Therefore, we can check if the factors are correct are not.

Note: While substituting the value of \[x\] in the equation, the substitution and the calculation should be done neatly and step wise. This method of checking helps us not to make mistakes and even if we had calculated wrongly, we can always redo it and correct it.