Question

factor. check by using foil.
1
$8x^{2}+2x - 3$

2
$6x^{2}-31x - 17$

3
$4x^{2}-20x - 39$

Explanation:

Problem 1: \(8x^2 + 2x - 3\)

Step 1: Find \(ac\) and factors

For \(ax^2+bx+c\), \(a = 8\), \(b=2\), \(c=-3\). \(ac=8\times(-3)=-24\). Find two numbers that multiply to \(-24\) and add to \(2\). The numbers are \(6\) and \(-4\).

Step 2: Split the middle term

Rewrite the middle term: \(8x^2 + 6x - 4x - 3\).

Step 3: Group and factor

Group: \((8x^2 + 6x) + (-4x - 3)\). Factor out GCF: \(2x(4x + 3) - 1(4x + 3)\). Then factor out \((4x + 3)\): \((4x + 3)(2x - 1)\).

Step 4: Check with FOIL

\((4x + 3)(2x - 1)=4x\times2x+4x\times(-1)+3\times2x+3\times(-1)=8x^2 - 4x + 6x - 3 = 8x^2 + 2x - 3\).

Step 1: Find \(ac\) and factors

\(a = 6\), \(b=-31\), \(c=-17\). \(ac=6\times(-17)=-102\). Find two numbers that multiply to \(-102\) and add to \(-31\). The numbers are \(-34\) and \(3\).

Step 2: Split the middle term

Rewrite the middle term: \(6x^2 + 3x - 34x - 17\).

Step 3: Group and factor

Group: \((6x^2 + 3x) + (-34x - 17)\). Factor out GCF: \(3x(2x + 1) - 17(2x + 1)\). Then factor out \((2x + 1)\): \((2x + 1)(3x - 17)\).

Step 4: Check with FOIL

\((2x + 1)(3x - 17)=2x\times3x+2x\times(-17)+1\times3x+1\times(-17)=6x^2 - 34x + 3x - 17 = 6x^2 - 31x - 17\).

Step 1: Find \(ac\) and factors

\(a = 4\), \(b=-20\), \(c=-39\). \(ac=4\times(-39)=-156\). Find two numbers that multiply to \(-156\) and add to \(-20\). The numbers are \(-26\) and \(6\).

Step 2: Split the middle term

Rewrite the middle term: \(4x^2 + 6x - 26x - 39\).

Step 3: Group and factor

Group: \((4x^2 + 6x) + (-26x - 39)\). Factor out GCF: \(2x(2x + 3) - 13(2x + 3)\). Then factor out \((2x + 3)\): \((2x + 3)(2x - 13)\).

Step 4: Check with FOIL

\((2x + 3)(2x - 13)=2x\times2x+2x\times(-13)+3\times2x+3\times(-13)=4x^2 - 26x + 6x - 39 = 4x^2 - 20x - 39\).

Answer:

\((4x + 3)(2x - 1)\)

Problem 2: \(6x^2 - 31x - 17\)