to factorize the expression a2 + 5ab - 84b2.

I will solve question 7(e) from the image.

The question asks to factorize the expression $a^2 + 5ab - 84b^2$.

Step 1: Identify the coefficients. This is a quadratic expression in terms of $a$ and $b$. We need to find two terms that multiply to the product of the first and last terms ($a^2 \times -84b^2 = -84a^2b^2$) and add up to the middle term ($5ab$).

Step 2: Find two numbers that multiply to $-84$ and add to $5$. We need two numbers whose product is $-84$ and whose sum is $5$. These numbers are $12$ and $-7$. So, the terms are $12ab$ and $-7ab$.

Step 3: Rewrite the middle term using these two terms. $a^2 + 5ab - 84b^2 = a^2 + 12ab - 7ab - 84b^2$

Step 4: Factor by grouping. Group the terms and factor out the common factors from each group. $(a^2 + 12ab) - (7ab + 84b^2)$ Factor $a$ from the first group and $7b$ from the second group. $a(a + 12b) - 7b(a + 12b)$

Step 5: Factor out the common binomial. $(a + 12b)(a - 7b)$

The factored form of $a^2 + 5ab - 84b^2$ is $(a + 12b)(a - 7b)$.

The final answer is $\boxed{(a + 12b)(a - 7b)}$.