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An algebraic expression is an expression built up from integer constants, variables, and the algebraic operations.

And, factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.

Let’s factorize the given algebraic expression.

We have a² + a – 132.

First: Multiply the coefficient of a² and the constant.

Coefficient of a² = 1

Constant = -132

⇒ 1 × -132 = -132 …(a)

Now, Observe the coefficient of a = 1

We need to split the value obtained at (a), such that the sum/difference of the split numbers comes out to be 1.

To split: We need to find the factors of -132.

Factors of -132 = 2, 2, 3, 11, -1

Add (2 × 2 × 3 =) 12 and (11 × -1 =) -11 = 12 + (-11) = 1

Multiply 12 and -11 = 12 × -11 = -132

Since, the sum of 12 and -11 is 1 and multiplication is -132. So, we can write

a² + a – 132 = a² + (12a – 11a) – 132

⇒ a² + a – 132 = a² + 12a – 11a – 132

⇒ a² + a – 132 = (a² + 12a) + (-11a – 132)

Now, take out common number or variable in first two pairs and the last two pairs subsequently.

⇒ a² + a – 132 = a(a + 12) – 11(a + 12)

Now, take out the common number or variable again from the two pairs.

⇒ a² + a – 132 = (a + 12)(a – 11)

Thus, the factorization of a² + a – 132 = (a + 12)(a – 11).