https://r2.erweima.ai/mathbot/image/17252160125770pw8e2z1.jpg

To solve the problems, we need to factorize each trinomial given in the image. The trinomials provided are:

1. \(8b^2 + 26b - 7\)
2. \(3m^2 - 7m - 20\)
3. \(6y^2 - 5y - 4\)

Let's factorize each trinomial step-by-step:

### 1. Factorize \(8b^2 + 26b - 7\)

To factorize the trinomial, we look for two numbers that multiply to \(8 \times (-7) = -56\) and add up to \(26\).

The two numbers are \(28\) and \(-2\).

Rewrite the middle term using these numbers:
\[
8b^2 + 28b - 2b - 7
\]

Group terms:
\[
(8b^2 + 28b) - (2b + 7)
\]

Factor each group:
\[
4b(2b + 7) - 1(2b + 7)
\]

Factor out the common term \((2b + 7)\):
\[
(4b - 1)(2b + 7)
\]

So, the factorization of \(8b^2 + 26b - 7\) is \((4b - 1)(2b + 7)\).

### 2. Factorize \(3m^2 - 7m - 20\)

We look for two numbers that multiply to \(3 \times (-20) = -60\) and add up to \(-7\).

The two numbers are \(5\) and \(-12\).

Rewrite the middle term:
\[
3m^2 - 12m + 5m - 20
\]

Group terms:
\[
(3m^2 - 12m) + (5m - 20)
\]

Factor each group:
\[
3m(m - 4) + 5(m - 4)
\]

Factor out the common term \((m - 4)\):
\[
(3m + 5)(m - 4)
\]

So, the factorization of \(3m^2 - 7m - 20\) is \((3m + 5)(m - 4)\).

### 3. Factorize \(6y^2 - 5y - 4\)

We look for two numbers that multiply to \(6 \times (-4) = -24\) and add up to \(-5\).

The two numbers are \(3\) and \(-8\).

Rewrite the middle term:
\[
6y^2 - 8y + 3y - 4
\]

Group terms:
\[
(6y^2 - 8y) + (3y - 4)
\]

Factor each group:
\[
2y(3y - 4) + 1(3y - 4)
\]

Factor out the common term \((3y - 4)\):
\[
(2y + 1)(3y - 4)
\]

So, the factorization of \(6y^2 - 5y - 4\) is \((2y + 1)(3y - 4)\).

### Summary of Factorizations

1. \(8b^2 + 26b - 7 = (4b - 1)(2b + 7)\)
2. \(3m^2 - 7m - 20 = (3m + 5)(m - 4)\)
3. \(6y^2 - 5y - 4 = (2y + 1)(3y - 4)\)

Would you like more details on any step, or do you have any questions?

### Related Questions:

1. How do you factor trinomials where the leading coefficient is not 1?
2. Can every trinomial be factored using integers?
3. What methods can be used to check if a trinomial is factorable?
4. How do you factor trinomials with complex coefficients?
5. What is the difference between factoring and solving a quadratic equation?

### Tip:

Always check your factorization by multiplying the factors to see if you get the original expression. This verifies the correctness of your factorization.

Aprende a factorizar trinomios paso a paso con ejemplos prácticos como 8b^2 + 26b - 7, 3m^2 - 7m - 20, y 6y^2 - 5y - 4. Encuentra soluciones detalladas y técnicas para factorizar polinomios con coeficientes no triviales.

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