**Sum Product Factoring** is a commonly used method of factoring on trinomials \( (ax^2 + bx + c)\) where the value of \(a\) is usually 1. In order to factor the trinomial properly, we need to find 2 integers whose **Product** is \(c\) and **Sum** is \(b\). These integers (better known as factors) will be referred to as \(r\) and \(s\). Once these values are found, create a factored expression in the form \((x+r)(x+s)\).

- Identify the values of \(b\) and \(c\) in the expression which determine the sum and product resepctively
- Create a table outlining the possible combinations of the sum and product based on the integer values
- Arange the values in a factored expression n the form \((x+r)(x+s)\)

Factor \(x^2+4x+3\)

We identify that \(b\) = 4 and \(c\) = 3; as a result, we need to calculate 2 integers whose \(\text{sum} = 4\) and \(\text{product} = 3\).

We can create a table that outlines the possible combinations for what the sum and product can be based on the 2 integer values. In this case, we try to identify what the sum is based on the product of \(3\):

r value **1**-1 s value **3**-3 Sum **4**-4 Product **3**3

Based on these results, we can determine that the \(r\) and \(s\) values (or factors) are \(1\) and \(3\) respectively.

We can arrange the values in a factored expression, \((x+1)(x+3)\).

Factor \(x^2+7x-18\)

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Enter in coefficients for the trinomial or click on the button to generate a new question. Try to factor the expression and then check your answer by clicking the "See Solution" button.

- Identify the values of \(a\), \(b\), and \(c\) in the expression where \(b\) determines the sum and \(ac\) determines the product
- Create a table outlining the possible combinations of the sum and product based on the integer values
- Once the \(r\) and \(s\) values are determined, rewrite the trinomial as a 4-term expression: \(ax^2+rx+sx+c\)
- Take the Greatest Common Factor (GCF) out of the 2 pairs of terms, \(ax^2+rx\) and \(sx+c\), to get common expressions
- Simplify by placing the GCF terms in its own expression

Factor \(2x^2+9x+10\)

We identify that \(a = 2\), \(b = 9\) and \(c = 10\); as a result, we need to calculate 2 integers whose \(\text{product} = 20\) and \(\text{sum} = 9\).

We can create a table that outlines the possible combinations for what the sum and product can be based on the 2 integer values. In this case, we try to identify what the sum is based on the product of \(20\):

r value 1 -1 2 -2 **4**-4 s value 20 -20 10 -10 **5**-5 Sum 21 -21 12 -12 **9**-9 Product 20 20 20 20 **20**20

Based on these results, we can determine that the \(r\) and \(s\) values (or factors) are \(4\) and \(5\) respectively.

We can arrange the values in a factored expression in the form \(ax^2+rx+sx+c\) which in this case will be \(2x^2+4x+5x+10\)

We can take out the GCF out of the 2 pairs of terms to get common expressions: \(2x(x+2)+5(x+2)\)

The GCF in the first expression was \(2x\), the GCF in the second expression was 5 and the common expression was \(x+2\)

Simplify by placing the GCF terms in its own expression: \((2x+5)(x+2)\)

Factor \(-3x^2-4x+20\)

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