Factorising the expression `9x^2 - 25x - 6` includes a coefficient for the `x^2` term (9).

This makes determining the factors for the quadratic more complicated.

The method is to

1. multiply the `x^2` coefficient and the integer;

2. determine the factors from the first step that add up to the `x` term coefficient;

3. rewrite the `x` term using these factors;

4. factorise the expression to get two brackets that are the same;

5. rewrite the expression with this new factorisation.

Factorise `8x^2 + 22x + 5`

(step 1) Multiply 8 and 5 to get 40

(step 2) Determine the factors of 40 that add to 22: 2 and 20

(step 3) Rewrite the expression as `8x^2 + 2x + 20x + 5`

(step 4) Factorise the expression `2x(4x + 1) + 5(4x + 1)`

(step 5) Write the expression in factorised form `(2x + 5)(4x + 1)`

(check) Multiply out the factorised expression to get `8x^2 + 2x + 20x + 5`

Answer: `(2x + 5)(4x + 1)`

Factorise `-21x^2 + 29x - 10`

(step 1) Multiply -21 and -10 to get 210

(step 2) Determine the factors of 210 that also add to 29: 14 and 15

(step 3) Rewrite the expression as `-21x^2 + 14x + 15x - 10`

(step 4) Factorise the expression `7x(-3x + 2) + 5(3x - 2)`

Adjust to make the terms in the brackets the same `7x(-3x + 2) - 5(-3x + 2)`

(step 5)Write the expression in factorised form `(7x - 5)(-3x + 2)`

(check) Multiply out to get `-21x^2 +15x + 14x -10`

Answer: `(-7x + 5)(3x - 2)`

See also Factorising Quadratic Expressions

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