What must be subtracted from \[3{a^2} - 6ab - 3{b^2} - 1\] to get \[4{a^2} - 7ab - 4{b^2} + 1\]\[?\]
A. \[ - {a^2} + ab + {b^3} - 2\]
B. \[ - {a^2} + ab + {b^2} - 2\]
C. \[{a^2} + ab + {b^2} - 2\]
D. \[{a^2} + ab + {b^3} - 2\]

Hint: we know that an algebraic expression is an expression which is made up of variables and constants, along with algebraic operations.
To find the expression which when subtracted from one given expression to get another given expression. We need to assume the unknown expression as a variable (or function). Then solving the equation obtained by combining the given expressions by using given statements.

Complete step by step answer:
Given \[3{a^2} - 6ab - 3{b^2} - 1\] ---(1)
And also given \[4{a^2} - 7ab - 4{b^2} + 1\] ---(2)
Suppose X be the expression such that when subtracted from the given expression (1) we get the expression (2).
Then,
\[3{a^2} - 6ab - 3{b^2} - 1 - X = 4{a^2} - 7ab - 4{b^2} + 1\] ---(3)
To obtained the value of X we need to express X separately in the equation (3), so we have to rewrite the equation (3) as follows
\[X = 3{a^2} - 6ab - 3{b^2} - 1 - (4{a^2} - 7ab - 4{b^2} + 1)\]----(4)
Simplifying the equation (4), we get
\[X = 3{a^2} - 6ab - 3{b^2} - 1 - 4{a^2} + 7ab + 4{b^2} - 1\]
\[ \Rightarrow \]\[X = - {a^2} + ab + {b^2} - 2\]

So, the correct answer is “Option B”.

Note:
In this type of questions, we have to know how to express the given statements correctly. Otherwise, the entire solution becomes wrong.
Also note that the expressions are made up of terms. The addition and subtraction of algebraic expressions can only be performed on like terms. The terms whose variables and their exponents are the same are known as like terms and the terms having different variables are unlike terms.
Additional information: There are four types of algebraic operations in mathematics. They are addition, subtraction, multiplication and division.